end, recall that the (continuous) random variable $U:=\Phi(X)\sim U(0,1)$ is
uniformly distributed on $[0,1]$, implying that $\Phi\in L^{2}(\gamma_{1})$,
with $\langle\Phi,1\rangle_{\gamma_{1}}=\mathbb{E}[U]=\frac{1}{2}$ and
$\|\Phi\|_{\gamma_{1}}^{2}=\mathbb{E}[U^{2}]=Var(U)+\mathbb{E}^{2}[U]=\frac{1}{12}+\frac{1}{4}=\frac{1}{3}$
(cf. [110, Remark 2.17.]). In particular, $\mathbb{E}[\kappa(X)]=0$, where
$\kappa:=2\sqrt{3}\Phi-\sqrt{3}=\sqrt{3}(2\Phi-1)$. Now, we are ready to prove

######  Proposition 6.23.

Let $X\sim N_{1}(0,1)$ and $\kappa:=\sqrt{3}\big{(}2\Phi-1\big{)}$. Then the
following properties hold:

  1. (i)

$\frac{\textup{d}^{n}}{\textup{d}t^{n}}\mathbb{E}[\Phi(X+t)]=2^{-n/2}\,\varphi^{(n-1)}\big{(}\frac{t}{\sqrt{2}}\big{)}\text{
for all }n\in\mathbb{N}\text{ and }t\in\mathbb{R}\,.$

  2. (ii)

$\mathbb{E}[\Phi(X+t)]=\Phi\big{(}\frac{t}{\sqrt{2}}\big{)}\text{ for all
}n\in\mathbb{N}\text{ and }t\in\mathbb{R}\,.$

  3. (iii)

$\mathbb{E}[\Phi(X)\,H_{2n+1}(X)]=(-1)^{n}\,\sqrt{\frac{1}{2\pi}}\,\sqrt{\frac{1}{4^{n}}\frac{1}{2n+1}{\binom{2n}{n}}}\,\sqrt{(\frac{1}{2})^{2n+1}}$
for all $n\in\mathbb{N}_{0}$. In particular,
$\mathbb{E}[\kappa(X)\,H_{2n+1}(X)]=(-1)^{n}\,\sqrt{\frac{6}{\pi}}\,\sqrt{\frac{1}{4^{n}}\frac{1}{2n+1}{\binom{2n}{n}}}\,\sqrt{(\frac{1}{2})^{2n+1}}$
for all $n\in\mathbb{N}_{0}$.

  4. (iv)

$h_{\Phi,\Phi}(\rho)=\frac{1}{4}+\frac{1}{2\pi}\,\arcsin(\frac{\rho}{2})\text{
and }h_{\kappa,\kappa}(\rho)=\frac{6}{\pi}\,\arcsin(\frac{\rho}{2})\text{ for
all }\rho\in[-1,1]$.

  5. (v)

$h_{\sqrt{3}\,\Phi,\sqrt{3}\,\Phi}=3\,h_{\Phi,\Phi}=\frac{3}{4}(1+\frac{2}{\pi}\,\arcsin(\frac{1}{2}\cdot))$
is a strictly increasing homeomorphic CCP function which maps $[-1,1]$ onto
$[\frac{1}{2},1]$ and is neither odd nor even. $h_{\kappa,\kappa}$ is an odd,
strictly increasing homeomorphic CCP function.
$h_{\sqrt{3}\,\Phi,\sqrt{3}\,\Phi}^{-1}(t)=-2\cos(\frac{2\pi}{3}t)$ for all
$t\in[\frac{1}{2},1]$, $h_{\kappa,\kappa}^{-1}(s)=2\sin(\frac{\pi}{6}s)$ and
$\big{(}h_{\kappa,\kappa}^{-1}\big{)}_{\text{abs}}(s)=2\sinh(\frac{\pi}{6}s)$
for all $s\in[-1,1]$. In particular,
$\big{(}h_{\kappa,\kappa}^{-1}\big{)}_{\text{abs}}(1)>1$.
$h_{\kappa,\kappa}^{\text{hyp}}(\rho)=\frac{6}{\pi}\sinh^{-1}(\frac{\rho}{2})$
for all $\rho\in[-1,1]$.

  6. (vi)

$\kappa$ is odd, $\|\kappa\|_{\gamma_{1}}=1$ and
$h_{\kappa,\kappa}^{\prime}(0)=\frac{3}{\pi}>0$. Moreover, $\kappa\in
L^{\infty}(\gamma_{1})$, and $\|\kappa\|_{\infty}=\sqrt{3}$.

###### Proof.

(i) Fix $t\in\mathbb{R}$. We prove this result by induction on $n$. Obviously,
we have to verify the base case only ($n=1$). To this end, we apply a few
Fourier transform techniques including the well-known fact that
$\widehat{\varphi}=\varphi$ is its own Fourier transform. So, let $n=1$. Then

$\frac{\textup{d}}{\textup{d}t}\mathbb{E}[\Phi(X+t)]=\int_{\mathbb{R}}(\tau_{-t}\varphi)(x)\widehat{\varphi}(x)\textup{d}x=\int_{\mathbb{R}}\widehat{(\tau_{-t}\varphi)}(x)\varphi(x)\textup{d}x\,,$

where $\tau_{-t}\varphi(x):=\varphi(x-(-t))$ denotes the translation operator
(here, applied to $\varphi$). Since
$\widehat{(\tau_{-t}\varphi)}(x)=\widehat{\varphi}(x)\exp(itx)=\varphi(x)\exp(itx)$
for all $x\in\mathbb{R}$, it follows that

$\displaystyle\frac{\textup{d}}{\textup{d}t}\mathbb{E}[\Phi(X+t)]$
$\displaystyle=\frac{1}{2\pi}\int_{\mathbb{R}}\exp(-x^{2})\exp(itx)\textup{d}x=\frac{1}{\sqrt{2}}\,\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\varphi(y)\exp\big{(}i\frac{t}{\sqrt{2}}y\big{)}\textup{d}y$
$\displaystyle=\frac{1}{\sqrt{2}}\widehat{\varphi}(-\frac{t}{\sqrt{2}})=\frac{1}{\sqrt{2}}\varphi(-\frac{t}{\sqrt{2}})=\frac{1}{\sqrt{2}}\varphi^{(0)}(\frac{t}{\sqrt{2}}),$

which finishes the verification of the base case.
(ii) We just have to integrate the base case on both sides (from $0$ to $t$,
say).
(iii) Since the Taylor series representation of $\varphi$ obviously is given
by

$\varphi(x)=\frac{1}{\sqrt{2\pi}}\exp(-\frac{x^{2}}{2})=\frac{1}{\sqrt{2\pi}}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{2^{m}\,m!}\,x^{2m}$

for all $x\in\mathbb{R}$, it follows that
$\varphi^{(2m)}(0)=(2m)!\big{(}\frac{1}{\sqrt{2\pi}}\frac{(-1)^{m}}{2^{m}\,m!}\big{)}=\frac{1}{\sqrt{2\pi}}\frac{(-1)^{m}\,m!}{2^{m}}\begin{pmatrix}2m\\\
m\end{pmatrix}$ and $\varphi^{(2m+1)}(0)=0$ for all $m\in\mathbb{N}_{0}$.
(iii) now follows directly from (i) and (the one-dimensional case of) the
equality (6.139), respectively Proposition 6.3.
(iv) follows directly from (iii).
(v) follows from (iv), Theorem 6.5, Lemma 6.11 and the fact that for example
$h_{\sqrt{3}\,\Phi,\sqrt{3}\,\Phi}(-1)=\frac{1}{2}\notin\\{-1,1\\}$.
(vi) Since $\Phi(x)+\Phi(-x)=1$ for all $x\in\mathbb{R}$, it follows that
$\kappa$ is odd. (iv), applied to $\rho=1$ implies that
$\|\kappa\|_{\gamma_{k}}^{2}=h_{\kappa,\kappa}(1)=1$. Moreover, (iv) clearly
implies that $h_{\kappa,\kappa}^{\prime}(0)=\frac{3}{\pi}>0$. Finally, since
$\Phi:\mathbb{R}\longrightarrow[0,1]$ is a distribution function, it follows
that $\kappa$ is bounded and
$|\kappa(x)|\leq\sqrt{3}=\sqrt{3}\lim\limits_{x\to\infty}(2\Phi(x)-1)=\lim\limits_{x\to\infty}\kappa(x)$
for all $x\in\mathbb{R}$. ∎

We will soon realise that Theorem 6.5 actually reflects a characterisation of
the class of all real continuous CCP functions (see Theorem 6.33).

Within the scope of our analysis of the Grothendieck constants, we need
invertible CCP functions, implying that we may ignore even CCP functions,
defined on $[-1,1]$ (since these are non-injective). However, thanks to
Theorem 5.17, $[-1,1]\ni\rho\mapsto\rho\,h(\rho)$ is an odd CCP function for
any even CCP function $h$. In fact, we will recognise next that in particular
any CCP function $h_{f,f}$, which satisfies $h_{f,f}(-1)<1$ induces
canonically an odd CCP function $h_{f^{\times},f^{\times}}$, where
$f^{\times}\in S_{H}$ is odd, $H:=L^{2}(\mathbb{R}^{k},\gamma_{k})$. In the
complex case it seems that we even have to use odd functions, to avoid
calculations with non-positive semidefinite entrywise absolute values of
correlation matrices (cf. (7.226) and Theorem 7.14, (ii)).

######  Proposition 6.24 (Odd CCP transform).

Let $k\in\mathbb{N}$ and $f\in L^{2}(\mathbb{R}^{k},\gamma_{k})$. Consider the
odd function $\mathbb{R}^{k}\ni x\mapsto
f^{\text{odd}}(x):=\frac{f(x)-f(-x)}{2}$ (the odd part of $f$). Then $f$ is
odd if and only if $f^{\text{odd}}=f$. In particular,
$(f^{\text{odd}})^{\text{odd}}=f^{\text{odd}}$. $f$ is even if and only if
$f^{\text{odd}}=0$. Moreover, $f^{\text{odd}}\in
L^{2}(\mathbb{R}^{k},\gamma_{k})$ and

  1. (i)

$\displaystyle\|f^{\text{odd}}\|_{\gamma_{k}}^{2}=\frac{h_{f,f}(1)-h_{f,f}(-1)}{2}\geq
0\,.$

  2. (ii)

$f$ is even $\lambda_{k}$-a.s. if and only if $h_{f,f}(-1)=h_{f,f}(1)$.

  3. (iii)

$f$ is odd $\lambda_{k}$-a.s. if and only if $h_{f,f}(-1)=-h_{f,f}(1)$.

  4. (iv)

Assume that $h_{f,f}(1)>h_{f,f}(-1)$. Put

$\displaystyle
f^{\times}:=\sqrt{\frac{2}{h_{f,f}(1)-h_{f,f}(-1)}}\,f^{\text{odd}}\,.$
(6.175)

Then $f$ is not even, $\|f^{\times}\|_{\gamma_{k}}=1$, and
$h_{f^{\times},f^{\times}}=\frac{2}{h_{f,f}(1)-h_{f,f}(-1)}\,h_{f^{\text{odd}},f^{\text{odd}}}$
is an odd CCP function.

###### Proof.

Due to Theorem 6.5, respectively Theorem 6.8, we just have to verify the non-
trivial parts of the claims (i), (ii) and (iii).
(i) Fix $\textbf{X}\sim N_{k}(0,I_{k})$. Then

$\|f^{\text{odd}}\|_{\gamma_{k}}^{2}=\frac{1}{4}\,\mathbb{E}[(f^{\circ})^{2}(\textbf{X})]=\frac{1}{4}\big{(}\mathbb{E}[f^{2}(\textbf{X})]-2\,\mathbb{E}[f(\textbf{X})f(\textbf{-X})]+\mathbb{E}[f^{2}(\textbf{-X})]\big{)}.$

Since $\textbf{-X}\stackrel{{\scriptstyle d}}{{=}}\textbf{X}\sim
N_{k}(0,I_{k})$ and ${\text{vec}}(\textbf{X},\textbf{-X})\sim
N_{2k}(0,\Sigma_{2k}(-1))$, it follows from Theorem 6.5 that

$\|f^{\text{odd}}\|_{\gamma_{k}}^{2}=\frac{1}{2}\big{(}\|f\|_{\gamma_{k}}^{2}-\mathbb{E}[f(\textbf{X})f(\textbf{-X})]\big{)}=\frac{1}{2}(h_{f,f}(1)-h_{f,f}(-1)).$

(ii) follows directly from (i).
(iii) Consider the even function $f^{\text{ev}}:=f-f^{\text{odd}}\in
L^{2}(\mathbb{R}^{k},\gamma_{k})$; i.e., the even part of $f$. By
construction, $f^{\text{ev}}(x)=\frac{f(x)+f(-x)}{2}$ for all
$x\in\mathbb{R}^{k}$. Similarly, as in the proof of (i), we therefore obtain

$\|f^{\text{ev}}\|_{\gamma_{k}}^{2}=\frac{h_{f,f}(1)+h_{f,f}(-1)}{2}\,,$

wherefrom claim (iii) obviously follows. ∎

######  Example 6.25.

Since $\arcsin(\frac{1}{2})=\frac{\pi}{6}$, Proposition 6.23-(iv) implies that
$h_{\Phi,\Phi}(1)=\frac{1}{3}$ and $h_{\Phi,\Phi}(-1)=\frac{1}{6}$, whence

$\Phi^{\times}=\sqrt{3}(2\Phi-1)=\kappa\,.$

######  Example 6.26.

Let $[-1,1]\ni\rho\mapsto C^{\text{Ga}}(\frac{1}{2},\frac{1}{2};\rho)$ denote
the bivariate Gaussian copula with Pearson’s correlation coefficient $\rho$ as
parameter, evaluated at $(\frac{1}{2},\frac{1}{2})$. Then

$C^{\text{Ga}}(\frac{1}{2},\frac{1}{2};\rho)=\Phi_{\Sigma_{2}(\rho)}(\Phi^{-1}(\frac{1}{2}),\Phi^{-1}(\frac{1}{2}))=\mathbb{P}(X\leq
0,Y\leq 0)=h_{g,g}(\rho)\,,$

where $\Phi_{\Sigma_{2}(\rho)}$ denotes the bivariate distribution function of
the random vector $(X,Y)^{\top}\sim N_{2}(0,\Sigma_{2}(\rho))$ and
$g:=11_{(-\infty,0]}$. Thus, $g^{\text{odd}}=-\frac{1}{2}\,\text{sign}$,
$h_{g,g}(1)=\mathbb{P}(X\leq 0)=\Phi(0)=\frac{1}{2}$ and
$h_{g,g}(-1)=\mathbb{P}(X\leq 0,-X\leq 0)=\mathbb{P}(X=0)=0$. Consequently,

$g^{\times}=-\text{sign}=2\,11_{(-\infty,0)}-1+11_{\\{0\\}}\,,$

and it follows that $h_{g^{\times},g^{\times}}=\frac{2}{\pi}\arcsin$ (on
$[-1,1]$).

Regarding an explicit calculation of the correlation coefficients
$\mathbb{E}[f(\textbf{X})g(\textbf{Y})]$, induced by a pair of non-linearly
transformed Gaussian random vectors and given “sufficiently smooth” functions
$f$ and $g$, we implement a few facts from the theory of distributions and
test function spaces. A detailed in-depth introduction to these “generalised
functions” and test function spaces and their analysis is provided by e. g.
[60, 75, 126]. To this end, let $k\in\mathbb{N}$, $f\in
L^{1}_{{\text{loc}}}(\mathbb{R}^{k})$ (i.e., $f$ is locally integrable) and
$\psi\in{\mathcal{S}}_{k}$ be an arbitrary test function, where
${\mathcal{S}}_{k}$ denotes the Schwartz space of rapidly decreasing test
functions on $\mathbb{R}^{k}$. Put

$\langle\psi,\Lambda_{f}\rangle:=\int_{\mathbb{R}^{k}}f(x)\psi(x)\textup{d}^{k}{x}\,.$

Observe that the latter symbol $\langle\cdot,\cdot\rangle$ denotes the duality
bracket on ${\mathcal{S}}_{k}\times{\mathcal{S}}_{k}^{\prime}$ and not an
inner product of Hilbert space elements. When there is no ambiguity, we adopt
the common habit to identify the tempered distribution
$\Lambda_{f}\in{\mathcal{S}}_{k}^{\prime}$ with $f\in
L^{1}_{{\text{loc}}}(\mathbb{R}^{k})$ itself. More generally, recall that
tempered distributions and their derivatives are elements of the dual space
${\mathcal{S}}_{k}^{\prime}$ of ${\mathcal{S}}_{k}$, such as the Dirac delta
distribution $\delta_{0}=\delta$, defined via
$\langle\psi,\delta\rangle:=\psi(0)$ for any $\psi\in{\mathcal{S}}_{k}$.
Observe that there is no $g\in L^{1}_{{\text{loc}}}(\mathbb{R}^{k})$ such that
$\delta=\Lambda_{g}$.

Let us quickly recall how differentiation of tempered distributions is
defined. It originates from a reiteration of the integration by parts formula
and is also known as “weak differentiation” (cf. e. g. [126, Chapter 6.12]):

$\displaystyle\langle\psi,D^{n}u\rangle:=(-1)^{|n|}\langle
D^{n}\psi,u\rangle=(-1)^{|n|}\langle\psi,D^{n-1}(Du)\rangle$ (6.176)

for all $\psi\in{\mathcal{S}}_{k},u\in{\mathcal{S}}_{k}^{\prime}$ and
$n\in\mathbb{N}_{0}^{k}$. Consequently, since
$H_{n}\,\varphi_{k}\in{\mathcal{S}}_{k}$ for all $n\in\mathbb{N}_{0}^{k}$ and
$D^{n}u\in{\mathcal{S}}_{k}^{\prime}$ for all $u\in{\mathcal{S}}_{k}$, (6.129)
in particular implies:

$\langle
H_{n}\,\varphi_{k},u\rangle=\frac{1}{\sqrt{n!}}\langle\varphi_{k},D^{n}u\rangle\text{
for all }u\in{\mathcal{S}}_{k}^{\prime}\text{ and }n\in\mathbb{N}_{0}^{k}\,.$

Given an arbitrary compact subset $K\subseteq\mathbb{R}^{k}$ and $f\in
L^{2}(\gamma_{k})$, it follows that

$\int_{K}|f|\,\textup{d}\lambda_{k}=\sqrt{2\pi}\int_{\mathbb{R}^{k}}11_{K}(x)\exp(\frac{1}{2}\|x\|^{2})\,|f(x)|\,\gamma_{k}(\textup{d}x)\leq
c_{K}\,\|f\|_{\gamma_{k}}\,,$

where
$c_{K}:=\sqrt{2\pi}\int_{K}e^{\|x\|^{2}}\,\gamma_{k}(\textup{d}x)=\sqrt{2\pi}\int_{K}\exp(\frac{1}{2}\|x\|^{2})\lambda_{k}(\textup{d}x)<\infty$.
Consequently, $L^{2}(\gamma_{k})\subseteq
L^{1}_{{\text{loc}}}(\mathbb{R}^{k})$, implying that for any $\textbf{X}\sim
N_{k}(0,I_{k})$

$\displaystyle\mathbb{E}[H_{n}(\textbf{X})f(\textbf{X})]=\langle
H_{n},f\rangle_{\gamma_{k}}=\langle
H_{n}\,\varphi_{k},\Lambda_{f}\rangle=\frac{1}{\sqrt{n!}}\langle\varphi_{k},D^{n}\Lambda_{f}\rangle\text{
for all }f\in L^{2}(\gamma_{k})\text{ and }n\in\mathbb{N}_{0}^{k}\,.$ (6.177)

######  Example 6.27 ($n$’th weak derivative of $\Lambda_{11_{[0,\infty)}}$).

Let $k=1$ and $n\in\mathbb{N}_{0}$. A very easy proof by induction on $n$,
based on (6.176), together with (6.128) firstly reveals that

$\langle\varphi,D^{n}\delta\rangle=(-1)^{n}\,\varphi^{(n)}(0)=\frac{\sqrt{n!}}{\sqrt{2\pi}}\,H_{n}(0)\,.$

(6.130) therefore implies that

$\langle\varphi,D^{2l}\delta\rangle=(-1)^{l}\,\frac{1}{\sqrt{2\pi}}\,(2l-1)!!\,\text{
for all }l\in\mathbb{N}_{0}\,.$

Similarly, (6.176) implies that $D\Lambda_{11_{[0,\infty)}}=\delta$. Since
$\text{sign}=2\,11_{[0,\infty)}-1$ ($\gamma_{1}$-almost surely) and $\langle
H_{2l+1},1\rangle_{\gamma_{1}}=\langle
H_{2l+1},H_{0}\rangle_{\gamma_{1}}=\delta_{2l+1,0}=0$, it follows that

$\displaystyle\langle H_{2l+1},\text{sign}\rangle_{\gamma_{1}}$
$\displaystyle=2\langle
H_{2l+1},11_{[0,\infty)}\rangle_{\gamma_{1}}\stackrel{{\scriptstyle\eqref{eq:Hermite_f_bracket}}}{{=}}\frac{2}{\sqrt{(2l+1)!}}\langle\varphi,D^{2l+1}\Lambda_{11_{[0,\infty)}}\rangle$
$\displaystyle=\frac{2}{\sqrt{(2l+1)!}}\langle\varphi,D^{2l}\delta\rangle=(-1)^{l}\,\sqrt{\frac{2}{\pi}}\,\frac{(2l-1)!!}{\sqrt{(2l+1)!}}$

for all $l\in\mathbb{N}_{0}$. This outcome – which is solely based on a
multiple weak differentiation of $\Lambda_{11_{[0,\infty)}}$ – should be
compared now with (the derivation of) (6.173) !

(6.177) can be strongly simplified if $f$ is smooth (cf. also Proposition
6.3). More precisely, if $N\in\mathbb{N}_{0},D^{n}f\in L^{2}(\gamma_{k})\cap
C(\mathbb{R}^{k})$ for all $n\in\mathbb{N}_{0}^{k}$, satisfying $0\leq|n|\leq
N$, and if $\textbf{X}\sim N_{k}(0,I_{k})$, then
$D^{n}\Lambda_{f}=\Lambda_{D^{n}f}$ (cf. [126, Chapter 6.13]), and

$\displaystyle\mathbb{E}[H_{n}(\textbf{X})f(\textbf{X})]=\langle
H_{n},f\rangle_{\gamma_{k}}=\frac{1}{\sqrt{n!}}\mathbb{E}[D^{n}f(\textbf{X})]\text{
for all }n\in\mathbb{N}_{0}^{k}\,,\text{ with }|n|\leq N\,.$ (6.178)

In fact, even more can be said. Recall that (by construction of Sobolev
spaces) the latter equality also holds without the smoothness assumption, if
we just assume that $f\in L^{2}(\gamma_{k})\cap
W_{{\text{loc}}}^{N,1}(\mathbb{R}^{k})$, so that in this case $D^{n}f$ denotes
the distributional $n$’th derivative in $L_{{\text{loc}}}^{1}(\mathbb{R}^{k})$
instead (cf. e. g. [60, Chapter 3.1]). Independent of an application of
Theorem 6.5 to the primary topic of our paper, it implies further non-trivial
consequences including a generalisation of Stein’s Lemma (cf. [91, Theorem 1
and Example 1]) and a certain “decorrelation property” of harmonic functions.
We only have to combine (6.178) and Theorem 6.5, resulting at once in

######  Proposition 6.28.

Let $k\in\mathbb{N}$, $\rho\in[-1,1]$ and
${\text{vec}}(\textbf{X},\textbf{Y})\sim N_{2k}(0,\Sigma_{2k}(\rho))$. Let
$f,g\in L^{2}(\gamma_{k})$. Then

${\text{cov}}(f(\textbf{X}),g(\textbf{Y}))=\sum_{\nu=1}^{\infty}\big{(}\sum_{n\in
C(\nu,k)}\frac{1}{n!}\,\langle\varphi_{k},D^{n}\Lambda_{f}\rangle\,\langle\varphi_{k},D^{n}\Lambda_{g}\rangle\big{)}\,\rho^{\nu}\,.$

If in addition $(f,g)\in W_{{\text{loc}}}^{N,1}(\mathbb{R}^{k})\times
W_{{\text{loc}}}^{N,1}(\mathbb{R}^{k})$, or if
$(D^{n}f,D^{n}g)\in(L^{2}(\gamma_{k})\cap
C(\mathbb{R}^{k}))\times(L^{2}(\gamma_{k})\cap C(\mathbb{R}^{k}))$ for all
$n\in\mathbb{N}_{0}^{k}$, satisfying $0\leq|n|\leq N$ for some
$N\in\mathbb{N}$, then

${\text{cov}}(f(\textbf{X}),g(\textbf{Y}))=\sum_{\nu=1}^{N}\big{(}\sum_{n\in
C(\nu,k)}\frac{1}{n!}\,\mathbb{E}[D^{n}f(\textbf{X})]\mathbb{E}[D^{n}g(\textbf{Y})]\big{)}\rho^{\nu}+\sum_{\nu=N+1}^{\infty}\big{(}\sum_{n\in
C(\nu,k)}\frac{1}{n!}\,\langle\varphi_{k},D^{n}\Lambda_{f}\rangle\,\langle\varphi_{k},D^{n}\Lambda_{g}\rangle\big{)}\,\rho^{\nu}\,.$

In particular, if $N=1$, then

${\text{cov}}(f(\textbf{X}),Y_{i})=\rho\,\mathbb{E}[\frac{\partial f}{\partial
x_{i}}(\textbf{X})]$

for all $i\in[k]$, respectively

$\mathbb{E}[f(\textbf{X})\textbf{Y}]=\rho\,\mathbb{E}[\nabla f(\textbf{X})].$

If $N=2$, then

${\text{cov}}(f(\textbf{X}),\|\textbf{Y}\|^{2})=\rho^{2}\,\mathbb{E}[\triangle
f(\textbf{X})].$

In the one-dimensional case (i. e., $k=1$), a direct application of
Proposition 6.28, respectively (6.178), implies two further results which are
of their own interest. In particular, our approach enables the provision of a
quick and short proof of the following generalisation of Stein’s Lemma to
standard Gaussian random powers (cf. [98, Theorem 1]):

######  Corollary 6.29.

Let $m\in\mathbb{N}_{0}$, $f\in L^{2}(\gamma_{1})$ and $X\sim N_{1}(0,1)$ be
given. If $D^{l}f\equiv f^{(l)}\in L^{2}(\gamma_{1})\cap C(\mathbb{R})$ for
all $0\leq l\leq m$, then

$\mathbb{E}[f(X)X^{m}]=\mathbb{E}[(-i)^{m}\,H_{m}(iD)(f(X))]=\sum_{\nu=0}^{\left\lfloor
m/2\right\rfloor}\binom{m}{2\nu}(2\nu-1)!!\,\mathbb{E}[D^{m-2\nu}f(X)]\,.$

###### Proof.

Fix $m\in\mathbb{N}_{0}$. Put $f_{m}(x):=x^{m}$ ($x\in\mathbb{R}$). Firstly,
recall the well-known result that

$\mathbb{E}[f_{\nu}(X)]=\mathbb{E}[X^{\nu}]=\frac{1+(-1)^{\nu}}{2}\,(\nu-1)!!\text{
for all }\nu\in\mathbb{N}_{0}\,,$

implying that

$\langle
f_{m},H_{l}\rangle_{\gamma_{1}}\stackrel{{\scriptstyle\eqref{eq:of_Stein_type}}}{{=}}\frac{1}{\sqrt{l!}}\,\mathbb{E}[D^{l}f_{m}(X)]=\begin{cases}\frac{1+(-1)^{m-l}}{2}\,\sqrt{l!}\,\binom{m}{l}\,(m-l-1)!!&\text{if
}l\leq m\\\ 0&\text{if }l>m\end{cases}$

Consequently,

$\mathbb{E}[f(X)f_{m}(X)]=h_{f,f_{m}}(1)=\sum_{l=0}^{m}\langle
f,H_{l}\rangle_{\gamma_{1}}\,\langle
f_{m},H_{l}\rangle_{\gamma_{1}}=\sum_{l=0}^{m}\frac{1+(-1)^{m-l}}{2}\mathbb{E}[D^{l}f(X)]\,\binom{m}{l}(m-l-1)!!\,.$

If $m$ is even, it therefore follows that

$\mathbb{E}[f(X)f_{m}(X)]=\sum_{k=0}^{m/2}\mathbb{E}[D^{2k}f(X)]\,\binom{m}{2k}(m-2k-1)!!=\sum_{\nu=0}^{m/2}\mathbb{E}[D^{m-2\nu}f(X)]\,\binom{m}{2\nu}(2\nu-1)!!\,.$

Similarly, if $m$ is odd, we obtain

$\mathbb{E}[f(X)f_{m}(X)]=\sum_{\nu=0}^{(m-1)/2}\mathbb{E}[D^{m-2\nu}f(X)]\,\binom{m}{2\nu}(2\nu-1)!!\,.$

However, since $\binom{m}{2\nu}(2\nu-1)!!=H_{m,\nu}$ (due to (6.128)), the
claim clearly follows. ∎

######  Corollary 6.30.

Let $\nu\in\mathbb{N}_{0}$, $f,g\in L^{2}(\gamma_{1})$ and $\rho\in(-1,1)$.
Then

$h_{f,g}^{(\nu)}(\rho)=\sum_{n=0}^{\infty}\frac{1}{n!}\,\langle\varphi,D^{n}(D^{\nu}\Lambda_{f})\rangle\langle\varphi,D^{n}(D^{\nu}\Lambda_{g})\rangle\rho^{n}\,.$

In particular, if in addition $(f,g)\in C^{\infty}(\mathbb{R}^{k})\times
C^{\infty}(\mathbb{R}^{k})$, then

$h_{f,g}^{(\nu)}(\rho)=h_{f^{(\nu)},g^{(\nu)}}(\rho)\text{ for all
}\rho\in(-1,1).$

###### Proof.

Since $h_{f,g}$ is real analytic, $h_{f,g}^{(\nu)}$ is well-defined (where
$h_{f,g}^{(0)}:=h_{f,g}$, of course). Due to (6.177) and Theorem 6.5, applied
to $k=1$, the remaining part of the proof is a straightforward proof by
induction on $\nu\in\mathbb{N}_{0}$. ∎

######  Remark 6.31.

Firstly, observe that the strong impact of the standard (multivariate)
Gaussian law, is reflected in (6.177), primarily implied by the fact that the
Gaussian density function $\varphi_{k}$ and hence each $H_{n}\,\varphi_{k}$,
is rapidly decreasing: $H_{n}\,\varphi_{k}\in{\mathcal{S}}_{k}$ for all
$n\in\mathbb{N}_{0}^{k}$ (since each $H_{n}$ is a polynomial). This fact and
the structure of the (multivariate) Hermite polynomials (which itself is also
induced by the structure of $\varphi_{k}$) namely enables a reiterated use of
the integration by parts formula in the smooth case, respectively a use of the
Sobolev space $W_{{\text{loc}}}^{N,1}(\mathbb{R}^{k})$ in the non-smooth case,
implying the transition of each inner product $\langle
H_{n},f\rangle_{\gamma_{k}}$ on the Hilbert space $L^{2}(\gamma_{k})$ into the
duality bracket $\langle\varphi_{k},D^{n}\,\Lambda_{f}\rangle$ on
${\mathcal{S}}_{k}\times{\mathcal{S}}_{k}^{\prime}$. The latter, however,
seems to be more convenient for performing specific computations.

###  6.4 Upper bounds of $K_{G}^{\mathbb{R}}$ and inversion of real CCP
functions

Our next step is to embed Grothendieck’s original approach as well as
Krivine’s improvement into a general framework. We are going to show that we
may substitute the Grothendieck function
$h_{\text{sign},\text{sign}}=\frac{2}{\pi}\arcsin$ through invertible CCP
functions $h_{f,f}$, generated by bounded functions
$f:\mathbb{R}^{k}\longrightarrow\mathbb{R}$ (see Theorem 6.40 and Theorem
6.45). Various proofs of Krivine’s main result
($K_{G}^{\mathbb{R}}\leq\frac{\pi}{2\ln(1+\sqrt{2})}\approx 1,782$) are set
out in detail in [69, Section 5], [76, proof of Lemma 10.5] and [150]. We will
recognise soon that inverses of odd CCP functions also lead to a further
crucial construction of correlation matrices, lurking in the following
important implication of Lemma 5.10:

######  Theorem 6.32.

Let $r>0$, $0<c\leq r$ and $0\not=\psi\in W^{\omega}_{+}((-r,r))$. Then
$\psi_{\text{abs}}\big{|}_{[0,r]}$ is strictly increasing and
$\psi_{\text{abs}}(c)>0$. For any $k\in\mathbb{N}$ there exist
$\alpha_{k}\equiv\alpha^{\psi,c}_{k}\in S_{L^{2}(\gamma_{k})}$ and
$\beta_{k}\equiv\beta^{\psi,c}_{k}\in S_{L^{2}(\gamma_{k})}$, such that the
following properties hold:

  1. (i)

If $\textbf{X}\sim N_{k}(0,I_{k})$, then
$\sqrt{\psi_{\text{abs}}(c)}\,\mathbb{E}[\alpha_{k}(\textbf{X})]=\text{sign}(\psi(0))\sqrt{|\psi(0)|}$
and
$\sqrt{\psi_{\text{abs}}(c)}\,\mathbb{E}[\beta_{k}(\textbf{X})]=\sqrt{|\psi(0)|}$.
In particular,
$\mathbb{E}[\alpha_{k}(\textbf{X})]=\text{sign}(\psi(0))\mathbb{E}[\beta_{k}(\textbf{X})]$.

  2. (ii)

If $c<r$, then $[-c,c]\subseteq(-r,r)$ and

$\displaystyle\psi(c\rho)=\psi_{\text{abs}}(c)h_{\alpha_{k},\beta_{k}}(\rho)\text{
for all }\rho\in[-1,1].$ (6.179)

In particular,
$\psi(c)=\psi_{\text{abs}}(c)\langle\alpha_{k},\beta_{k}\rangle_{\gamma_{k}}$.

  3. (iii)

If $c\leq r$, then

$\psi_{\text{abs}}(c\rho)=\psi_{\text{abs}}(c)\,h_{\alpha_{k},\alpha_{k}}(\rho)=\psi_{\text{abs}}(c)\,h_{\beta_{k},\beta_{k}}(\rho)\text{
for all }\rho\in[-1,1].$

  4. (iv)

If $c<r$, $H$ is an arbitrary $\mathbb{R}$-Hilbert space, then there exists a
$\mathbb{R}$-Hilbert space $\mathbb{H}$ such that for any $u,v\in S_{H}$

$\displaystyle\psi(c\langle
u,v\rangle_{H})=\psi_{\text{abs}}(c)\langle\psi_{u}(\alpha_{k}),\psi_{v}(\beta_{k})\rangle_{\mathbb{H}}=\langle
a_{u},b_{v}\rangle_{\mathbb{H}}\,,$ (6.180)

where for any $w\in S_{H}$,
$\|a_{w}\|_{\mathbb{H}}^{2}=\|b_{w}\|_{\mathbb{H}}^{2}=\psi_{\text{abs}}(c)$
and $\psi_{w}:L^{2}(\gamma_{k})\longrightarrow\mathbb{H}$ is a mapping which
satisfies $\psi_{w}(S_{L^{2}(\gamma_{k})})\subseteq S_{\mathbb{H}}$. In
particular, $\psi(c)=\langle a_{w},b_{w}\rangle_{\mathbb{H}}$ for all $w\in
S_{H}$.
If $c\leq r$, then

$\displaystyle\psi_{\text{abs}}(c\langle u,v\rangle_{H})$
$\displaystyle=\psi_{\text{abs}}(c)\langle\psi_{u}(\alpha_{k}),\psi_{v}(\alpha_{k})\rangle_{\mathbb{H}}=\psi_{\text{abs}}(c)\langle\psi_{u}(\beta_{k}),\psi_{v}(\beta_{k})\rangle_{\mathbb{H}}$
$\displaystyle=\langle a_{u},a_{v}\rangle_{\mathbb{H}}=\langle
b_{u},b_{v}\rangle_{\mathbb{H}}\,,$

###### Proof.

Let $0<c\leq r$. Put $b_{\nu}:=\frac{\psi^{(\nu)}(0)}{\nu!}$
($\nu\in\mathbb{N}_{0}$). Consider the family
$\\{q_{n}:n\in\mathbb{N}_{0}^{k}\\}$, defined as
$q_{0}:=\sqrt{|\psi(0)|}=\sqrt{|b_{0}|}$ and

$q_{n}\equiv
q_{n}(k):=\sqrt{\frac{|\psi^{(|n|)}(0)|}{n!\,k^{|n|}}}=\sqrt{\frac{|n|\,!\,|b_{|n|}|}{n!\,k^{|n|}}}=\sqrt{\frac{|n|\,!}{n!}\,a_{|n|}^{\,|n|}}=\sqrt{\frac{|n|\,!}{n!}\,\prod\limits_{i=1}^{k}a_{|n|}^{n_{i}}}$

for $n\in\mathbb{N}_{0}^{k}\setminus\\{0\\}$, where $a_{\nu}\equiv
a_{\nu}(k):=\big{(}\frac{|b_{\nu}|}{k^{\nu}}\big{)}^{1/\nu}$ ($\nu\not=0$). As
before, let $C(\nu;k):=\\{n\in\mathbb{N}_{0}^{k}:|n|=\nu\\}$, where
$\nu\in\mathbb{N}_{0}$. A straightforward application of the multinomial
theorem implies that for all $\nu\in\mathbb{N}$ the following equality holds:

$\displaystyle\sum_{n\in C(\nu;k)}q_{n}^{2}(k)=\sum_{n\in
C(\nu;k)}\frac{\nu!}{n!}\,\prod\limits_{i=1}^{k}a_{\nu}^{n_{i}}=\sum_{n\in
C(\nu;k)}\binom{\nu}{n_{1},n_{2},\ldots,n_{k}}\,\prod\limits_{i=1}^{k}a_{\nu}^{n_{i}}=\big{(}\sum_{i=1}^{k}a_{\nu}\big{)}^{\nu}=k^{\nu}a_{\nu}^{\nu}=|b_{\nu}|\,.$
(6.181)

It should be noted here, that by construction also $\sum\limits_{n\in
C(0;k)}q_{n}^{2}=q_{0}^{2}=|b_{0}|$ holds. Consequently, since
$\\{H_{\alpha}:\alpha\in\mathbb{N}_{0}^{k}\\}$ is an orthonormal basis in
$L^{2}(\gamma_{k})$, it follows that

$\alpha^{\psi,c}_{k}:=\frac{1}{\sqrt{\psi_{\text{abs}}(c)}}\sum_{n\in\mathbb{N}_{0}^{k}}\text{sign}(b_{|n|})\,q_{n}(k)(\sqrt{c})^{|n|}H_{n}\in
S_{L^{2}(\gamma_{k})}$

and

$\beta^{\psi,c}_{k}:=\frac{1}{\sqrt{\psi_{\text{abs}}(c)}}\sum_{n\in\mathbb{N}_{0}^{k}}q_{n}(k)(\sqrt{c})^{|n|}H_{n}\in
S_{L^{2}(\gamma_{k})}$

are well-defined (including the case $c=r$), from which the assertions (i),
(ii) and (iii) follow (due to (6.181)). (iv) follows from (ii), (iii) and
Proposition 6.4, with $\rho:=\langle u,v\rangle_{H}$, and (v) is an
implication of Lemma 5.10 and the construction of $\alpha_{k}$ and
$\beta_{k}$. ∎

If we combine Theorem 5.17, Theorem 6.5, Proposition 6.24 and Theorem 6.32,
(iii), a further significant characterisation of continuous odd CCP functions
follows at once:

######  Theorem 6.33 (CCP representation theorem).

Let $\psi:[-1,1]\longrightarrow\mathbb{R}$ be a continuous odd function and
$k\in\mathbb{N}$. Then the following statements are equivalent:

  1. (i)

$\psi$ is a CCP function.

  2. (ii)

$\psi=h_{f,f}$ for some odd $f\in S_{L^{2}(\gamma_{k})}$.

Theorem 6.32-(iv), together with Lemma 6.11 directly enriches us with another
crucial result, related to a construction of real quantum correlation matrices
which actually include lurking upper bounds of $K_{G}^{\mathbb{R}}$:

######  Corollary 6.34.

Let $m,n,k\in\mathbb{N}$ and $f\in S_{L^{2}(\gamma_{k})}$ be odd. Assume that
$h_{f,f}^{-1}{\big{|}}_{(-1,1)}\in W^{\omega}_{+}((-1,1))$. Put

$c(f):=h_{f,f}^{\text{hyp}}(1)\,.$

Then $c(f)\in(0,1)$ and

$h_{f,f}^{-1}[c(f)S]\in{\mathcal{Q}}_{m,n}\,\text{ for all
}\,S\in{\mathcal{Q}}_{m,n}\,.$

If $\Sigma=\begin{pmatrix}M&S\\\ S^{\top}&N\end{pmatrix}\in C(m+n;\mathbb{R})$
is an arbitrary real $(m+n)\times(m+n)$ correlation matrix (with block
elements $M\in C(m;\mathbb{R}),N\in C(n;\mathbb{R})$ and
$S\in{\mathcal{Q}}_{m,n}$), then

$\displaystyle\begin{pmatrix}(h_{f,f}^{-1})_{\text{abs}}[c(f)M]&h_{f,f}^{-1}[c(f)S]\\\\[6.99997pt]
h_{f,f}^{-1}[c(f)S^{\top}]&(h_{f,f}^{-1})_{\text{abs}}[c(f)N]\end{pmatrix}\in
C(m+n;\mathbb{R})$ (6.182)

again is an $(m+n)\times(m+n)$ correlation matrix with real entries.

###### Proof.

We just have to apply Theorem 6.32, (iv), including (6.180), to the function
$\psi:=h_{f,f}^{-1}{\big{|}}_{(-1,1)}$ and the constant
$c:=h_{f,f}^{\text{hyp}}(1)$ ($0<c<1$, due to Lemma 6.11). ∎

If $\psi:D_{\mathbb{F}}\longrightarrow D_{\mathbb{F}}$ is an arbitrary CCP
function and $\Sigma\in C(m+n;\mathbb{F})$ is an arbitrary $m+n$-correlation
matrix, then

$\psi[\Sigma]=\begin{pmatrix}\psi[\Gamma_{H}(u,u)]&\psi[\Gamma_{H}(u,v)]\\\\[1.99997pt]
\psi[\Gamma_{H}(u,v)^{\ast}]&\psi[\Gamma_{H}(v,v)]\end{pmatrix}\in
C(m+n;\mathbb{F})$

again is a correlation matrix. Consequently, it follows that
$\psi[\Gamma_{H}(u,u)]=\Gamma_{K}(x,x)\in C(m;\mathbb{F})$,
$\psi[\Gamma_{H}(v,v)]=\Gamma_{K}(y,y)\in C(n;\mathbb{F})$ and
$\psi[\Gamma_{H}(u,v)]=\Gamma_{K}(x,y)\in\mathcal{Q}_{m,n}(\mathbb{F})$ for
some Hilbert space $K$ over $\mathbb{F}$ and some $(x,y)\in S_{K}^{m}\times
S_{K}^{n}$ (due to Corollary 3.5). However, a considerably stronger result
holds (cf. also (6.182)):

######  Theorem 6.35.

Let $m,n\in\mathbb{N}$ and $M:=\begin{pmatrix}A&S\\\
S^{\top}&B\end{pmatrix}\in\mathbb{M}_{m+n}(\mathbb{R})$, where
$A\in\mathbb{M}_{m}(\mathbb{R})$, $S\in\mathbb{M}_{m,n}(\mathbb{R})$ and
$B\in\mathbb{M}_{n}(\mathbb{R})$. Let $0<r<\infty$ and
$f,g:(-r,r)\longrightarrow\mathbb{R}$ be two functions, such that $(-r,r)\ni
x\mapsto f(x)=\sum_{\nu=0}^{\infty}a_{\nu}x^{\nu}\in W^{\omega}_{+}((-r,r))$
and $(-r,r)\ni x\mapsto g(x)=\sum_{\nu=0}^{\infty}b_{\nu}x^{\nu}\in
W^{\omega}_{+}((-r,r))\in W^{\omega}_{+}((-r,r))$. Let $q\in
C^{\omega}((-r,r))$, such that

$\displaystyle|c_{\nu}|\leq\sqrt{|a_{\nu}|\,|b_{\nu}|}\text{ for all
}\nu\in\mathbb{N}_{0}\,,$ (6.183)

where $c_{\nu}:=\frac{q^{(\nu)}(0)}{\nu!}$. Then $q\in
W^{\omega}_{+}((-r,r))$, and the following properties hold:

  1. (i)

If $M\in\mathbb{M}_{m+n}([-r,r])^{+}$ is positive semidefinite, and if any
entry of the matrices $A,S$ and $B$ is an element of $[-r,r]$, then also

$\begin{pmatrix}f_{\text{abs}}[A]&\widetilde{q}[S]\\\
\widetilde{q}[S]^{\top}&g_{\text{abs}}[B]\end{pmatrix}\in\mathbb{M}_{m+n}([-r,r])^{+}$

is positive semidefinite, where $\widetilde{q}$ is defined as in Lemma 5.10.
In particular, if $0<c^{\ast}\leq r$ is a root of $f_{\text{abs}}-1$, then

$\widetilde{q}[c^{\ast}\Gamma]\in\mathcal{Q}_{m,n}\text{ for all
}\Gamma\in\mathcal{Q}_{m,n}\,.$

  2. (ii)

If $M\in C(m+n;\mathbb{R})$ is a real correlation matrix, then also

$\begin{pmatrix}h_{\alpha,\alpha}[A]&h_{\alpha,\beta}[S]\\\
h_{\alpha,\beta}[S^{\top}]&h_{\beta,\beta}[B]\end{pmatrix}\in
C(m+n;\mathbb{R})$

is a real correlation matrix for all $\alpha,\beta\in S_{L^{2}(\gamma_{k})}$.
In particular,

$\displaystyle
h_{\alpha,\beta}[\cdot]:{\mathcal{Q}}_{m,n}\longrightarrow{\mathcal{Q}}_{m,n}\text{
for all }\alpha,\beta\in S_{L^{2}(\gamma_{k})}.$ (6.184)

  3. (iii)

For all $\alpha,\beta\in S_{L^{2}(\gamma_{k})}$, for all Hilbert spaces $H$
and $u,v\in S_{H}$ there exist $d\in\mathbb{N}$ and $x,y\in\mathbb{S}^{d-1}$,
such that

$h_{\alpha,\beta}(\langle u,v\rangle_{H})=\langle
x,y\rangle_{\mathbb{R}_{2}^{d}}=x^{\top}y\,.$

###### Proof.

(i) Our proof is built on a sufficient and necessary characterisation of
positive semidefinite block matrices, Schur multiplication of matrices, and a
straightforward application of Hölder’s inequality in $l_{2}$. Due to [55,
Lemma 1.1.13, statement 2], it suffices to show that the following inequality
is satisfied and well-defined:

$\displaystyle|h^{\top}q[S]k|\leq\sqrt{(h^{\top}f_{\text{abs}}[A]h)}\sqrt{(k^{\top}g_{\text{abs}}[B]k)}\text{
for all }(h,k)\in\mathbb{R}^{m}\times\mathbb{R}^{n}\,.$ (6.185)

So, fix $h\in\mathbb{R}^{m}$ and $k\in\mathbb{R}^{n}$. Given the assumption
that $M$ is positive semidefinite, it follows that also $A$ and $B$ are
positive semidefinite matrices. Since
$M^{\ast\,0}={\bf{1}}_{m+n}{\bf{1}}_{m+n}^{\top}$,
$A^{\ast\,0}={\bf{1}}_{m}{\bf{1}}_{m}^{\top}$,
$B^{\ast\,0}={\bf{1}}_{n}{\bf{1}}_{n}^{\top}$ and
$S^{\ast\,0}={\bf{1}}_{m}{\bf{1}}_{n}^{\top}$, the Schur multiplication
theorem therefore implies that in fact for any $\nu\in\mathbb{N}_{0}$

$M^{\ast\,\nu}=\begin{pmatrix}A^{\ast\,\nu}&S^{\ast\,\nu}\\\
\big{(}S^{\ast\,\nu}\big{)}^{\top}&B^{\ast\,\nu}\end{pmatrix}$

is positive semidefinite, such as $A^{\ast\,\nu}$ and $B^{\ast\,\nu}$. Hence,

$|h^{\top}S^{\ast\,\nu}k|\leq\sqrt{h^{\top}A^{\ast\,\nu}h}\,\sqrt{k^{\top}B^{\ast\,\nu}k}\text{
for all }\nu\in\mathbb{N}_{0}$

(due to [55, Lemma 1.1.13.2]). Put
$\alpha_{\nu}:=\sqrt{|a_{\nu}|}\,\sqrt{h^{\top}(A^{\ast\,\nu})h}=\sqrt{h^{\top}(|a_{\nu}|A^{\ast\,\nu})h}$
and
$\beta_{\nu}:=\sqrt{|b_{\nu}|}\,\sqrt{h^{\top}(B^{\ast\,\nu})h}=\sqrt{k^{\top}(|b_{\nu}|B^{\ast\,\nu})k}$.
An application of Hölder’s inequality in $l_{2}$ implies that

$\displaystyle\sum_{\nu=0}^{\infty}\alpha_{\nu}\beta_{\nu}\leq\big{(}\sum_{\nu=0}^{\infty}\alpha_{\nu}^{2}\big{)}^{\frac{1}{2}}\,\big{(}\sum_{\nu=0}^{\infty}\beta_{\nu}^{2}\big{)}^{\frac{1}{2}}=\sqrt{(h^{\top}f_{\text{abs}}[A]h)}\sqrt{(k^{\top}g_{\text{abs}}[B]k)}.$
(6.186)

Due to (7.230), a further application of Hölder’s inequality in $l_{2}$
implies that

$\sum_{\nu=0}^{\infty}|c_{\nu}|\,r^{\nu}\stackrel{{\scriptstyle\eqref{eq:T0_C_omega_inequality}}}{{\leq}}\sum_{\nu=0}^{\infty}\sqrt{|a_{\nu}|\,r^{\nu}}\,\sqrt{|b_{\nu}|\,r^{\nu}}\leq\big{(}\sum_{\nu=0}^{\infty}|a_{\nu}|\,r^{\nu}\big{)}^{\frac{1}{2}}\big{(}\sum_{\nu=0}^{\infty}|b_{\nu}|\,r^{\nu}\big{)}^{\frac{1}{2}}=\sqrt{f_{\text{abs}}(r)\,g_{\text{abs}}(r)}<\infty\,.$

Thus, $q\in W^{\omega}_{+}((-r,r))$. Hence,

$\displaystyle|h^{\top}\widetilde{q}[S]\,k|$
$\displaystyle=\big{|}h^{\top}\big{(}\sum_{\nu=0}^{\infty}c_{\nu}\,S^{\ast\,\nu}\big{)}k\big{|}\leq\sum_{\nu=0}^{\infty}|c_{\nu}||h^{\top}S^{\ast\,\nu}k|$
$\displaystyle\leq\sum_{\nu=0}^{\infty}\sqrt{|a_{\nu}|\,|b_{\nu}|}\,\sqrt{h^{\top}A^{\ast\,\nu}h}\,\sqrt{k^{\top}B^{\ast\,\nu}k}=\sum_{\nu=0}^{\infty}\alpha_{\nu}\,\beta_{\nu}\,.$

Thus, (6.186) implies (6.185). Suppose that $f_{\text{abs}}(c^{\ast})=1$ for
some $0<c^{\ast}\leq r$. Let $S\in\mathcal{Q}_{m,n}$. Due to Corollary 3.5, we
may assume without loss of generality that $A\in C(m;\mathbb{R})$ and $B\in
C(n;\mathbb{R})$, implying that $M\in C(m+n;\mathbb{R})$. In particular,
$c^{\ast}M=\begin{pmatrix}c^{\ast}A&c^{\ast}S\\\
(c^{\ast}S)^{\top}&c^{\ast}B\end{pmatrix}$ is positive semidefinite. Lemma
3.2-(ii) therefore implies that

${}_{f,g}M_{q}:=\begin{pmatrix}f_{\text{abs}}[c^{\ast}A]&\widetilde{q}[c^{\ast}S]\\\
\widetilde{q}[c^{\ast}S]^{\top}&g_{\text{abs}}[c^{\ast}B]\end{pmatrix}$

is already a correlation matrix (since
$f_{\text{abs}}(c^{\ast}a_{ii})=f_{\text{abs}}(c^{\ast})=1=f_{\text{abs}}(c^{\ast}b_{ii}))$.
Corollary 3.5 therefore concludes the proof of (i).
(ii) Put $f:=h_{\alpha,\alpha},g:=h_{\beta,\beta}$ and $q:=h_{\alpha,\beta}$.
Observe that $h_{\alpha,\beta}=q=\widetilde{q}\big{|}_{[-1,1]}$ (since
$h_{\alpha,\beta}$ is continuous). Firstly, since $\alpha,\beta\in
S_{L^{2}(\gamma_{k})}$, Theorem 6.5-(i) implies that $f(1)=1=g(1)$, so that
the diagonal of the matrix ${}_{f,g}M_{q}$ is occupied with $m+n$ $1$’s. It
remains to prove that ${}_{f,g}M_{q}$ is positive semidefinite. Recall that
$|p_{\nu}(\alpha,\beta)|\leq\sqrt{p_{\nu}(\alpha,\alpha)}\sqrt{p_{\nu}(\beta,\beta)}$
for all $\nu\in\mathbb{N}_{0}$, where $p_{\nu}(\psi,\kappa):=\sum_{n\in
C(\nu,k)}\langle\psi,H_{n}\rangle_{\gamma_{k}}\,\langle\kappa,H_{n}\rangle_{\gamma_{k}}=\frac{h_{\psi,\kappa}^{(\nu)}(0)}{\nu!},\psi,\kappa\in\\{\alpha,\beta\\}$
(cf. (6.141)). Theorem 6.5 and Theorem 6.33 therefore imply that all
assumptions of (i) are satisfied for the so defined functions $f,g$ and $q$
(including (7.230)), so that we may apply statement (i) also to these
functions. Finally, (6.184) and hence (iii) follows from Corollary 3.5 and
Proposition 3.13. ∎

However, we will recognise that Theorem 6.35 cannot be fully transferred to
the complex field, so that we have to distinguish carefully between the real
case and the complex case here (cf. Proposition 7.7). If we link (5.124) and
Theorem 6.35, we obtain

######  Corollary 6.36.

Let $m,n\in\mathbb{N},A\in\mathbb{M}_{m,n}(\mathbb{R})$ and

$\Sigma=\begin{pmatrix}M&S\\\ S^{\top}&N\end{pmatrix}\in C(m+n;\mathbb{R})$

be an arbitrary real $(m+n)\times(m+n)$ correlation matrix (with block
elements $M\in C(m;\mathbb{R}),N\in C(n;\mathbb{R})$ and
$S\in{\mathcal{Q}}_{m,n}$). Let $r>0$ and $0\not=\psi\in
W^{\omega}_{+}((-r,r))$. If $0<c\leq r$, then

$\displaystyle\frac{1}{\psi_{\text{abs}}(c)}\begin{pmatrix}\psi_{\text{abs}}[c\,M]&\widetilde{\psi}[c\,S]\\\\[5.0pt]
\widetilde{\psi}[c\,S]^{\top}&\psi_{\text{abs}}[c\,N]\end{pmatrix}\in
C(m+n;\mathbb{R})$ (6.187)

again is a correlation matrix with real entries. In particular,

$\displaystyle\frac{1}{\psi_{\text{abs}}(c)}\widetilde{\psi}[cS]\in{\mathcal{Q}}_{m,n}\,\text{
for all }\,S\in{\mathcal{Q}}_{m,n}\,.$ (6.188)

If we apply Theorem 6.32, (iv) and (6.188) to the inverses of invertible
functions in $W^{\omega}_{+}((-1,1))$, Bolzano’s intermediate value theorem
from calculus immediately implies a further crucial result. To this end,
recall also (5.123) and (5.124).

######  Theorem 6.37 (Real inner product rounding).

Let $\psi:[-1,1]\longrightarrow[-1,1]$ be a bijective real function. Assume
that

$\psi{|}_{(-1,1)}\in W^{\omega}_{+}((-1,1))\text{ and
}\psi^{-1}{|}_{(-1,1)}\in W^{\omega}_{+}((-1,1)).$

Then
$|\psi^{-1}(0)|=(\psi^{-1}{|}_{(-1,1)})_{\text{abs}}(0)\leq(\psi^{-1}{|}_{(-1,1)})_{\text{abs}}(1)$.
Assume that

$|\psi^{-1}(0)|<1<(\psi^{-1}{|}_{(-1,1)})_{\text{abs}}(1).$

Then there is a unique number $c^{\ast}\in(0,1)$, such that
$(\psi^{-1})_{\text{abs}}(c^{\ast})=1$. Let $k\in\mathbb{N}$. There is
$(\alpha,\beta)\in S_{L^{2}(\gamma_{k})}\times S_{L^{2}(\gamma_{k})}$
(dependent on $c^{\ast}$ and $k$) such that for any separable
$\mathbb{R}$-Hilbert space $H$ and any $u,v\in S_{H}$, the following
statements apply:

  1. (i)

$\displaystyle\langle u,v\rangle_{H}=\frac{1}{c^{\ast}}\psi(\rho_{u,v})$
(6.189)

where $\rho_{u,v}:=h_{\alpha,\beta}(\langle u,v\rangle_{H})$.

  2. (ii)

$c^{\ast}=\psi(\langle\alpha,\beta\rangle_{\gamma_{1}}).$

  3. (iii)

Suppose that $\psi=h_{f,g}$ for some $\nu\in\mathbb{N}$ and $f,g\in
L^{2}(\gamma_{\nu})$. If $\rho_{u,v}\in(-1,1)$, then

$c^{\ast}\,\langle
u,v\rangle_{H}=\frac{1}{(2\pi)^{\nu}(1-\rho_{u,v}^{2})^{\nu/2}}\,\int_{\mathbb{R}^{\nu}}\int_{\mathbb{R}^{\nu}}f(x)g(y)\exp\big{(}-\frac{\|x\|^{2}+\|y\|^{2}-2\rho_{u,v}\langle
x,y\rangle_{2}}{2(1-\rho_{u,v}^{2})}\big{)}\textup{d}^{\nu}x\,\textup{d}^{\nu}y\,.$

If $m,n\in\mathbb{N}$ and $(x,y)\in S_{H}^{m}\times S_{H}^{n}$, then there
exist $m+n$ $\mathbb{R}^{k}$-valued random vectors
$\textbf{X}_{1},\ldots,\textbf{X}_{m},\textbf{Y}_{1},\ldots,\textbf{Y}_{n}$,
such that ${\text{vec}}(\textbf{X}_{i},\textbf{Y}_{j})\sim
N_{2k}(0,\Sigma_{2k}(\rho_{ij}))$ for all $(i,j)\in[m]\times[n]$, and

$\displaystyle\Gamma_{H}(x,y)=\frac{1}{c^{\ast}}h_{f,g}[R]=\frac{1}{c^{\ast}}\,\mathbb{E}[{\textbf{P}}_{f}{\textbf{Q}}_{g}^{\top}]=\frac{1}{c^{\ast}}\,\mathbb{E}[\Gamma_{\mathbb{R}}({\textbf{P}}_{f},{\textbf{Q}}_{g})],$
(6.190)

where $({\textbf{P}}_{f})_{i}:=f(\textbf{X}_{i})$,
$({\textbf{Q}}_{g})_{j}:=g(\textbf{Y}_{j})$,
$\rho_{ij}:=h_{\alpha,\beta}(\langle x_{i},y_{j}\rangle_{H})$ and
$R:=h_{\alpha,\beta}[\Gamma_{H}(x,y)]=(\rho_{ij})_{(i,j)\in[m]\times[n]}\in\mathcal{Q}_{m,n}$.

  4. (iv)

Moreover, $c^{\ast}\langle u,v\rangle_{H}\in(-1,1)$ for all $u,v\in S_{H}$ and

$\displaystyle\text{tr}(A^{\top}S)=\frac{1}{c^{\ast}}\,\text{tr}(A^{\top}\,\psi[Q_{c^{\ast},\psi}])$
(6.191)

for all $m,n\in\mathbb{N}$, for all $A\in\mathbb{M}_{m,n}(\mathbb{R})$, for
all $S\in\mathcal{Q}_{m,n}$, where
$Q_{c^{\ast},\psi}:=\psi^{-1}[c^{\ast}\,S]\in\mathcal{Q}_{m,n}$.

It is far from being trivial that it is possible to transfer Theorem 6.37 from
the real field $\mathbb{R}$ to the complex field $\mathbb{C}$; at least if
$f=g$ is odd (cf. Theorem 7.13). In order to achieve this, we have to develop
and implement certain non-trivial structural properties of the class of
complex Hermite polynomials (cf. Theorem 7.11 and Theorem 7.12 below).

If $\nu\in\mathbb{N}_{0}$, $m,n\in\mathbb{N}$ and
$A\in\mathbb{M}_{m,n}(\mathbb{F})$, then $A^{\ast\,\nu}$ denotes the $\nu$-th
entrywise power of the matrix $A$ (in terms of Schur multiplication), where
$A^{\ast\,0}:={\bf{1}}_{m}{\bf{1}}_{n}^{\ast}$ is the (rank 1) $m\times n$
matrix of all ones, where
${\bf{1}}_{l}:=(1,1,\ldots,1)^{\top}\in\mathbb{F}^{l},l\in\mathbb{N}$ (by
adapting the convention that $0^{0}:=1$ \- cf. [81, Remark 9.2]).

Now, we are fully prepared to embed both, Grothendieck’s original estimation
$K_{G}^{\mathbb{R}}\leq\sinh(\frac{\pi}{2})\approx 2.301$, and Krivine’s
original estimation $K_{G}^{\mathbb{R}}\leq\frac{\pi}{2\ln(1+\sqrt{2})}\approx
1,782$ into a general framework. In particular, we will provide a further
proof of the Grothendieck inequality itself (see Theorem 6.40 below).
Moreover, we will make a cute use of the little Grothendieck inequality,
yielding a quite surprising outcome. To this end, we have to work with
functions $h_{f,g}$, which are “generated” by bounded functions $f,g\in
L^{\infty}(\mathbb{R}^{k},\gamma_{k})=L^{\infty}(\mathbb{R}^{k},\lambda_{k})\equiv
L^{\infty}(\mathbb{R}^{k})$, $k\in\mathbb{N}$. (6.146) obviously implies that

$|h_{f,g}(\rho)|\leq\|f\|_{\infty}\,\|g\|_{\infty}\,\text{ for all }\,f,g\in
L^{\infty}(\mathbb{R}^{k})\,\text{ and }\,\rho\in[-1,1]\,.$

######  Remark 6.38.

In statistical machine learning a function
$f:\mathbb{R}^{k}\longrightarrow\\{-1,1\\},k\in\mathbb{N}$ is a particular
example of a mapping from some general domain of definition to the set
$\\{-1,1\\}$, where the latter is known as “concept” (cf. e.g. [137]). The
concept $f$ obviously satisfies the condition
$\|f\|_{\gamma_{k}}=1=\|f\|_{\infty}$.

We also need the following important result which holds for both fields
interchangeably.

######  Lemma 6.39.

Let $\mathbb{F}\in\\{\mathbb{R},\mathbb{C}\\}$ and $m,n\in\mathbb{N}$. Let R
be a $m$-dimensional random vector in
$(\mathbb{F}\cap\overline{\mathbb{D}})^{m}$ and S be a $n$-dimensional random
vector in $(\mathbb{F}\cap\overline{\mathbb{D}})^{n}$, such that
$\mathbb{E}[{\textbf{R}}{\textbf{S}}^{\ast}]$ exists. Then

$\|B\ast\mathbb{E}[{\textbf{R}}{\textbf{S}}^{\ast}]^{\ast\nu}\|^{\mathbb{F}}_{\infty,1}\leq\|B\|^{\mathbb{F}}_{\infty,1}\text{
for all }B\in\mathbb{M}_{m,n}(\mathbb{F})\text{ and }\nu\in\mathbb{N}\,.$

###### Proof.

Clearly, it suffices to verify the statement for $\nu=1$. Without loss of
generality, we may assume that $m=n$ (just to simplify the spelling). So, fix
$B\in\mathbb{M}_{n}(\mathbb{F})$. Put
${\mathbf{\Lambda}}:={\textbf{R}}{\textbf{S}}^{\ast}=\Gamma_{\mathbb{F}}(\overline{\textbf{R}},\overline{\textbf{S}})$.
Because of the structure of the operator norm
$\|\cdot\|^{\mathbb{F}}_{\infty,1}$ and Lemma 5.6, it follows that

$\|B\ast\mathbb{E}[{\mathbf{\Lambda}}]\|^{\mathbb{F}}_{\infty,1}=|{\text{tr}}((B\ast\mathbb{E}[{\mathbf{\Lambda}}])^{\ast}\,p\,q^{\ast})|=|{\text{tr}}(B^{\ast}\,\mathbb{E}[(p\,q^{\ast})\ast\overline{{\mathbf{\Lambda}}}])|$

for some non-random vectors $p,q\in S_{\mathbb{F}}^{n}$. Given the assumed
structure of the random matrix ${\mathbf{\Lambda}}$, it follows that
$\overline{{\mathbf{\Lambda}}}=\overline{\textbf{R}}\,{\overline{\textbf{S}}}^{\ast}$.
Consequently,

$(p\,q^{\ast})\ast\overline{{\mathbf{\Lambda}}}=(p\,q^{\ast})\ast(\overline{\textbf{R}}\,{\overline{\textbf{S}}}^{\ast})=(p\ast\overline{\textbf{R}})(q\ast\overline{\textbf{S}})^{\ast}=\Gamma_{\mathbb{F}}(\overline{p}\ast\textbf{R},\overline{q}\ast\textbf{S})$

is a random Gram matrix, such that both random vectors
$\overline{p}\ast\textbf{R}$, and $\overline{q}\ast\textbf{S}$ map into
$(\mathbb{F}\cap\overline{\mathbb{D}})^{n}$. Hence,

$\|B\ast\mathbb{E}[{\mathbf{\Lambda}}]\|^{\mathbb{F}}_{\infty,1}=\big{|}{\text{tr}}(\mathbb{E}\big{[}B^{\ast}\,\Gamma_{\mathbb{F}}(\overline{p}\ast\textbf{R},\overline{q}\ast\textbf{S})\big{]})\big{|}\leq\mathbb{E}\big{[}\big{|}{\text{tr}}(B^{\ast}\,\Gamma_{\mathbb{F}}(\overline{p}\ast\textbf{R},\overline{q}\ast\textbf{S}))\big{|}\big{]}\leq\|B\|_{\infty,1}.$

∎

######  Theorem 6.40.

Let $k,m,n\in\mathbb{N}$ and $f,g\in L^{\infty}(\mathbb{R}^{k})$. Put
$r_{\infty}\equiv r_{\infty}(f,g):=\|f\|_{\infty}\,\|g\|_{\infty}$. Then:

  1. (i)

$\displaystyle|{\text{tr}}(A^{\top}h_{f,g}[S])|\leq
r_{\infty}\,\|A\|_{\infty,1}\text{ for all
}A\in\mathbb{M}_{m,n}(\mathbb{R})\text{ and }S\in{\mathcal{Q}}_{m,n}$ (6.192)

and

$\displaystyle|{\text{tr}}(A^{\top}h_{f,f}[\Sigma])|\leq\|f\|_{\infty}^{2}\,\max\limits_{x\in[-1,1]^{n}}|x^{\top}Ax|\text{
for all }A\in\mathbb{M}_{n}(\mathbb{R})\text{ and }\Sigma\in C(n;\mathbb{R}).$
(6.193)

  2. (ii)

Assume that $r_{\infty}>0$ and $f$ or $g$ is odd. Let
$h_{f,g}:[-1,1]\longrightarrow\mathbb{R}$ be continuous and injective. Then
$h_{f,g}$ is a homeomorphism which is either strictly increasing or strictly
decreasing and satisfies $h_{f,g}([-1,1])=[-r,r]$, where $r\equiv
r_{k}(f,g):=\max\\{-h_{f,g}(1),h_{f,g}(1)\\}=\max\\{-\langle
f,g\rangle_{\gamma_{k}},\langle f,g\rangle_{\gamma_{k}}\\}$. Moreover,
$0<r\leq r_{\infty}$. Assume that $h_{f,g}^{-1}{\big{|}}_{(-r,r)}\in
W^{\omega}_{+}((-r,r))$. Then the following two statements hold:

    1. (ii-1)

If $\big{(}h_{f,g}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(r)>1$, then
there is exactly one number $0<c^{\ast}_{k}\equiv c^{\ast}_{k}(f,g)<r$, such
that
$\big{(}h_{f,g}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(c^{\ast}_{k})=1$
and

$\displaystyle K_{G}^{\mathbb{R}}\leq\frac{r_{\infty}}{c^{\ast}_{k}}\,.$
(6.194)

    2. (ii-2)

If $r=r_{\infty}$, then there exists a unique number
$0<\gamma^{\ast}_{k}\equiv\gamma^{\ast}_{k}(f,g)\in(0,r]$, such that

$\displaystyle
K_{G}^{\mathbb{R}}=\big{(}h_{f,g}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(\gamma^{\ast}_{k})\leq\min\Big{\\{}\frac{r}{c^{\ast}_{k}}\,,\big{(}h_{f,g}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(r)\Big{\\}}\,.$
(6.195)

###### Proof.

Let $r_{\infty}>0$. Put $f^{\circ}:=\frac{f}{\|f\|_{\infty}}$ and
$g^{\circ}:=\frac{g}{\|g\|_{\infty}}$. If
$h_{f,g}:[-1,1]\longrightarrow[-a,a]$ is a well-defined and bijective, then
also
$h_{f^{\circ},g^{\circ}}=\frac{1}{r_{\infty}}\,h_{f,g}:[-1,1]\longrightarrow[-\tfrac{a}{r_{\infty}},\tfrac{a}{r_{\infty}}]$
exists and is bijective. Since
$h_{f^{\circ},g^{\circ}}^{-1}(\tfrac{y}{r_{\infty}})=h_{f,g}^{-1}(y)$ for all
$y\in[-a,a]$ and
$\max\\{-h_{f,g}(1),h_{f,g}(1)\\}=r_{\infty}\,\max\\{-h_{f^{\circ},g^{\circ}}(1),h_{f^{\circ},g^{\circ}}(1)\\}$,
we may assume throughout the entire proof without loss of generality that
$\|f\|_{\infty}=1$ and $\|g\|_{\infty}=1$, implying that $r_{\infty}=1$.
Fix $A\in\mathbb{M}_{m,n}(\mathbb{R})$ and $S\in{\mathcal{Q}}_{m,n}$. Then
$S=\Gamma_{H}(u,v)$ for some real finite-dimensional Hilbert space $H$ and
some $(u,v)\in S_{H}^{m}\times S_{H}^{n}$ (due to Proposition 3.13).
(i) Because of (6.155), it follows that

$\displaystyle
h_{f,g}[S]=h_{f,g}[\Gamma_{H}(u,v)]=\mathbb{E}[{\textbf{R}}_{f}\,{\textbf{T}}_{g}^{\top}]$
(6.196)

where $({\textbf{R}}_{f})_{i}:=f({\textbf{X}}_{u_{i}})$ and
$({\textbf{T}}_{g})_{j}:=g({\textbf{X}}_{v_{j}})$, for some
${\text{vec}}({\textbf{X}}_{u_{i}},{\textbf{X}}_{v_{j}})\sim
N_{2k}(0,\Sigma_{2k}(\langle u_{i},v_{j}\rangle_{H}))$ for all
$(i,j)\in[m]\times[n]$. Since $\|f\|_{\infty}=1=\|g\|_{\infty}$, the
stochastic inequality

$\displaystyle|{\text{tr}}(A^{\top}\,{\textbf{R}}_{f}\,{\textbf{T}}_{g}^{\top})|\leq\|A\|_{\infty,1}$
(6.197)

is satisfied (due to (3.41) and (3.42)). Hence,

$\displaystyle|{\text{tr}}(A^{\top}\,h_{f,g}[S])|$
$\displaystyle=|{\text{tr}}(A^{\top}\,\mathbb{E}[{\textbf{R}}_{f}\,{\textbf{T}}_{g}^{\top}])|=|\mathbb{E}[{\text{tr}}(A^{\top}\,{\textbf{R}}_{f}\,{\textbf{T}}_{g}^{\top})]|$
$\displaystyle\leq\,\,\,\mathbb{E}[|{\text{tr}}(A^{\top}\,{\textbf{R}}_{f}\,{\textbf{T}}_{g}^{\top})|]\stackrel{{\scriptstyle\eqref{eq:at_the_right_side_of_GT}}}{{\leq}}\|A\|_{\infty,1}\,.$

Similarly, by making use of (6.156), the proof of (6.193) can be performed
straightforwardly, so that (i) follows.
(ii) Firstly, since $h_{f,g}$ is odd, it follows that
$-h_{f,g}(1)=h_{f,g}(-1)$. If $h_{f,g}$ is strictly decreasing, then
$h_{f,g}(1)<h_{f,g}(-1)=-h_{f,g}(1)$, implying that $r=-h_{f,g}(1)=-\langle
f,g\rangle_{\gamma_{k}}>-h_{f,g}(0)=0$. Consequently,
$0<r=|h_{f,g}(1)|\leq\|f\|_{\gamma_{k}}\,\|g\|_{\gamma_{k}}\leq r_{\infty}$
(due to (6.145), respectively (6.146)). A similar proof obviously holds if
$h_{f,g}$ is strictly increasing. All described topological properties of
$h_{f,g}$ are a direct implication of well-known facts from classical real
analysis.
(ii-1) Assume that
$\big{(}h_{f,g}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(r)>1$. Since
$h_{f,g}^{-1}(0)=0=\big{(}h_{f,g}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(0)$,
we may apply Theorem 6.37, (iv) to $\psi:=\tfrac{1}{r}\,h_{f,g}$ (since
$\big{(}\psi^{-1}{\big{|}}_{(-1,1)}\big{)}_{\text{abs}}=\big{(}h_{f,g}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(r\,\cdot)$).
It follows that there exists a uniquely determined number
$0<\gamma_{k}^{\ast}\equiv\gamma_{k}^{\ast}(f,g)<1$ such that
$\big{(}\psi^{-1}{\big{|}}_{(-1,1)}\big{)}_{\text{abs}}(\gamma_{k}^{\ast})=1$
and
${\text{tr}}(A^{\top}S)=\tfrac{1}{\gamma_{k}^{\ast}}\,{\text{tr}}(A^{\top}\psi[S_{\ast}])=\tfrac{1}{r\,\gamma_{k}^{\ast}}\,{\text{tr}}(A^{\top}h_{f,g}[S_{\ast}])$
for some $S_{\ast}\in\mathcal{Q}_{m,n}$. Put
$0<c_{k}^{\ast}:=r\,\gamma_{k}^{\ast}<r$. Consequently, we may apply statement
(i), and it follows that

$\displaystyle|{\text{tr}}(A^{\top}S)|=\frac{1}{c_{k}^{\ast}}|{\text{tr}}(A^{\top}h_{f,g}[S_{\ast}])|\stackrel{{\scriptstyle\eqref{eq:property_of_quantum_corr_matrices}}}{{\leq}}\frac{1}{c_{k}^{\ast}}\,\|A\|_{\infty,1}\,.$
(6.198)

Proposition 3.15 therefore implies (6.194).
(ii-2) Assume that $r\geq r_{\infty}=1$. Due to the existence of the inverse
function $h_{f,g}^{-1}:[-r,r]\longrightarrow[-1,1]$, (6.196) implies that
(algebraically)

$S=h_{f,g}^{-1}[h_{f,g}[S]]=h_{f,g}^{-1}[\mathbb{E}[{\bf{\Lambda}}_{f,g}]],$

where ${\bf{\Lambda}}_{f,g}:={\textbf{R}}_{f}\,{\textbf{T}}_{g}^{\top}$ Put
$a_{\nu}:=\frac{(h_{f,g}^{-1})^{(\nu)}(0)}{\nu!}$, $\nu\in\mathbb{N}$. Since
by assumption $r\geq 1$, it follows that
$\sum_{\nu=0}^{\infty}|a_{\nu}|\leq\sum_{\nu=0}^{\infty}|a_{\nu}|r^{\nu}=\big{(}h_{f,g}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(r)<\infty$.
Lemma 5.10, applied to the real analytic function
$h_{f,g}^{-1}{\big{|}}_{(-r,r)}$ and the continuity of the function
$h_{f,g}^{-1}:[-r,r]\longrightarrow[-1,1]$ therefore imply that

$S=\sum_{\nu=0}^{\infty}a_{\nu}\,\mathbb{E}[{\bf{\Lambda}}_{f,g}]^{\ast\,\nu}\,,$

where $\ast$ denotes entrywise multiplication on $M_{m,n}(\mathbb{R})$ (i.e.,
the Hadamard product of matrices). Since $\|f\|_{\infty}=\|g\|_{\infty}=1$, it
follows that

$\displaystyle\|A_{m}\|_{\infty,1}\leq\|A\|_{\infty,1}\text{ for all
}m\in\mathbb{N}_{0},$ (6.199)

where $A_{m}:=A\ast\mathbb{E}[{\bf{\Lambda}}_{f,g}]^{\ast\,m}\hskip
1.42262pt(m\in\mathbb{N}_{0})$ (due to Lemma 6.39). Therefore, Lemma 5.6 leads
to

$\displaystyle|{\text{tr}}(A^{\top}\,S)|$
$\displaystyle\leq\sum_{\nu=0}^{\infty}|a_{\nu}|\,|{\text{tr}}(A^{\top}\,\mathbb{E}[{\bf{\Lambda}}_{f,g}]^{\ast\,\nu})|=|a_{0}||{\text{tr}}(A^{\top}{\bf{1}}_{m}{\bf{1}}_{n}^{\ast})|+\sum_{\nu=1}^{\infty}|a_{\nu}|\,|{\text{tr}}(A_{\nu-1}^{\top}\,\mathbb{E}[{\bf{\Lambda}}_{f,g}])|$
$\displaystyle\leq|a_{0}||{\text{tr}}(A^{\top}{\bf{1}}_{m}{\bf{1}}_{n}^{\ast})|+\sum_{\nu=1}^{\infty}|a_{\nu}|\,\mathbb{E}[|{\text{tr}}(A_{\nu-1}^{\top}\,{\bf{\Lambda}}_{f,g})|]$
$\displaystyle\\!\\!\\!\stackrel{{\scriptstyle\eqref{eq:at_the_right_side_of_GT}}}{{\leq}}|a_{0}|\|A\|_{\infty,1}+\sum_{\nu=1}^{\infty}|a_{\nu}|\,\|A_{\nu-1}\|_{\infty,1}\stackrel{{\scriptstyle\eqref{eq:Schur_product_estimation}}}{{\leq}}\big{(}\sum_{\nu=0}^{\infty}|a_{\nu}|\big{)}\|A\|_{\infty,1}$
$\displaystyle\leq\big{(}h_{f,g}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(r)\|A\|_{\infty,1}.$

Consequently, Proposition 3.15 implies that
$1<K_{G}^{\mathbb{R}}\leq\big{(}h_{f,g}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(r)$.
(6.195) now follows from (6.194) and Bolzano’s intermediate value theorem,
where the latter is applied to the strictly increasing function
$\big{(}h_{f,g}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}-K_{G}^{\mathbb{R}}$
(cf. proof of Theorem 6.8-(iii)). ∎

Inequality (6.193), together with the equality (3.47) allows us to recover as
special case (applied to $f=\text{sign}$) an interesting result of Y.
Nesterov, yet without having to make use of their random hyperplane rounding
technique (cf. [105]):

######  Corollary 6.41.

Let $k,n\in\mathbb{N}$, $A\in\mathbb{M}_{n}(\mathbb{R})^{+}$ be positive
semidefinite and $f\in L^{\infty}(\mathbb{R}^{k})$ be odd. Then

$\displaystyle h_{f,f}^{\prime}(0)\sup\limits_{\Sigma\in
C(n;\mathbb{R})}{\text{tr}}(A\Sigma)$ $\displaystyle\leq\sup\limits_{\Sigma\in
C(n;\mathbb{R})}{\text{tr}}(A\,h_{f,f}[\Sigma])\leq\|f\|_{\infty}^{2}\,\sup\limits_{x\in\\{-1,1\\}^{n}}x^{\top}Ax\leq\|f\|_{\infty}^{2}\,\|A\|_{\infty,1}\,.$

In particular,

$\displaystyle 0\leq
h_{f,f}^{\prime}(0)\leq\frac{2}{\pi}\,\|f\|_{\infty}^{2}\,.$ (6.200)

###### Proof.

Let $\Sigma\in C(n;\mathbb{R})$. Put $h\equiv h_{f,f}$. Since $f$ is odd,
Theorem 6.5-(v) implies that

$h(\rho)=\sum_{\nu=0}^{\infty}b_{2\nu+1}\rho^{2\nu+1}=h^{\prime}(0)\rho+\psi(\rho),$

for all $\rho\in[-1,1]$, where
$\psi(\rho):=\sum_{\nu=1}^{\infty}b_{2\nu+1}\rho^{2\nu+1}$ and $b_{2\nu+1}\geq
0$ for all $\nu\in\mathbb{N}_{0}$. Hence,
$\psi[\Sigma]=h[\Sigma]-h^{\prime}(0)\Sigma\in\mathbb{M}_{n}(\mathbb{R})^{+}$
(due to Theorem 5.15), whence
${\text{tr}}(\psi[\Sigma])={\text{tr}}(A(h[\Sigma])-h^{\prime}(0)\,{\text{tr}}(\Sigma))\geq
0$ (since ${\text{tr}}(AB)=\|B^{1/2}A^{1/2}\|_{F}^{2}\geq 0$ for all
$B\in\mathbb{M}_{n}(\mathbb{R})^{+}$). Thus,

$h^{\prime}(0)\,{\text{tr}}(A\Sigma)\leq{\text{tr}}(A\,h[\Sigma])\stackrel{{\scriptstyle\eqref{eq:property_of_corr_matrices}}}{{\leq}}\|f\|_{\infty}^{2}\,\sup\limits_{x\in\\{-1,1\\}^{n}}x^{\top}Ax=\|f\|_{\infty}^{2}\,\sup\limits_{x\in\\{-1,1\\}^{n}}{\text{tr}}(Axx^{\top})\leq\|f\|_{\infty}^{2}\,\|A\|_{\infty,1}.$

In order to verify (6.200), we are going to make use of the little
Grothendieck inequality. Obviously, we may assume that $h^{\prime}(0)>0$,
implying that

${\text{tr}}(B\Sigma)\leq\frac{1}{h^{\prime}(0)}\,\|f\|_{\infty}^{2}\,\|B\|_{\infty,1}\text{
for all }B\in\mathbb{M}_{n}(\mathbb{R})^{+}\text{ and }\Sigma\in
C(n;\mathbb{R}).$

Consequently, the “correlation matrix version” of the little Grothendieck
inequality (see (3.49)) implies

$\frac{\pi}{2}=k_{G}^{\mathbb{R}}\leq\frac{1}{h^{\prime}(0)}\,\|f\|_{\infty}^{2}\,,$

and (6.200) follows. ∎

In a similar vain, we directly obtain a further (and very short) proof for the
value of the real little Grothendieck constant $k_{G}^{\mathbb{R}}$. To this
end, we only have to combine (6.192) and Proposition 6.16:

######  Corollary 6.42 (Grothendieck, 1953).

$k_{G}^{\mathbb{R}}=\frac{\pi}{2}.$

###### Proof.

Let $A\in\mathbb{M}_{n}(\mathbb{R})^{+}$ and $\Sigma\in C(n;\mathbb{R})$ be
arbitrary. We just have to work with the CCP function $[-1,1]\ni\rho\mapsto
h_{f_{1},f_{1}}(\rho)-c_{1}^{2}\,\rho=\frac{2}{\pi}\arcsin(\rho)-\frac{2}{\pi}\,\rho$.
Because then

$0\leq\tfrac{2}{\pi}\,\text{tr}(A\Sigma)=\text{tr}(A(\tfrac{2}{\pi}\,\Sigma))\leq\text{tr}(Ah_{f_{1},f_{1}}[\Sigma])\stackrel{{\scriptstyle\eqref{eq:property_of_quantum_corr_matrices}}}{{\leq}}\|A\|_{\infty,1}\,.$

∎

######  Remark 6.43.

A direct estimation of (6.157) leads to the upper bound
$k\,\|f\|_{\infty}^{2}\,(\sqrt{\frac{2}{\pi}})^{2}=\frac{2k}{\pi}\,\|f\|_{\infty}^{2}$,
which, however, strongly depends on the dimension $k\in\mathbb{N}$. That upper
bound, viewed as a function of $k$ is even strictly increasing. Our
application of the little Grothendieck inequality implies the non-trivial
result that for all $k\in\mathbb{N}$, $h_{f,f}^{\prime}(0)$ actually is
bounded above by $\frac{2}{\pi}\|f\|_{\infty}^{2}$ “uniformly”. Moreover, if
$f$ were an even function, then (6.200) would be trivial, since
$h_{f,f}^{\prime}(0)=0$.

Theorem 6.40-(i) also implies a remarkable property of CCP functions. To this
end, let $S\in\mathcal{Q}_{m,n}$ and $h=h_{f,f}$ be an arbitrary CCP function
(not necessarily odd !), generated by some $f\in L^{\infty}(\mathbb{R}^{k})$.
Firstly, note that in any case,

$\frac{\pi}{2}\frac{1}{K_{G}^{\mathbb{R}}}\stackrel{{\scriptstyle\eqref{eq:known_upper_bounds_of_the_complex_GT_constant}}}{{<}}1=\|f\|_{\gamma_{k}}\leq\|f\|_{\infty}\leq\|f\|^{2}_{\infty}\,.$

If also $h_{f,f}^{-1}$ were a CCP function, then
$\widetilde{S}:=h_{f,f}^{-1}[S]\in\mathcal{Q}_{m,n}$ (due to (6.184)). Hence,

$|{\text{tr}}(A^{\top}S)|=|{\text{tr}}(A^{\top}h_{f,f}[\widetilde{S}])|\leq\|f\|^{2}_{\infty}\,\|A\|_{\infty,1}\text{
for all }A\in\mathbb{M}_{m,n}(\mathbb{R}),$

implying that $K_{G}^{\mathbb{R}}\leq\|f\|^{2}_{\infty}$. If we join the
latter observation and Theorem 5.17-(v), we obtain another interesting fact
(which should be carefully compared with Corollary 6.10):

######  Remark 6.44.

Let $k,m,n\in\mathbb{R}$ and $h=h_{f,f}$ be CCP for some $f\in
S_{L^{2}(\gamma_{k})}\cap L^{\infty}(\mathbb{R}^{k})$. If
$\|f\|_{\infty}<\sqrt{K_{G}^{\mathbb{R}}}$, then the inverse function
$h_{f,f}^{-1}$ is not CCP.

Next, we are going to summarise the remarkable properties of odd CCP functions
which we found so far and shed some light on an additional, quite surprising
estimate for odd CCP functions $h_{f,f}$ which emerges if we assume in
addition that $f\in S_{L^{2}(\gamma_{k})}$ is bounded (a.s.); i.e., if $f\in
S_{L^{2}(\gamma_{k})}\cap L^{\infty}(\mathbb{R}^{k})$. In particular, if we
combine Theorem 6.8-(ii), (v) and Theorem 6.40-(ii) in this case, we recover
Grothendieck’s upper bound as well as Krivine’s upper bound at once. This
follows from Example 6.48, which is a special case of our following key result
for the real odd CCP case:

######  Theorem 6.45.

Let $k\in\mathbb{N}$ and $f\in S_{L^{2}(\gamma_{k})}\cap
L^{\infty}(\mathbb{R}^{k})$ such that $\|f\|_{\gamma_{k}}=1$. Then
$\|f\|_{\infty}\geq 1$. Assume that $f$ is odd and
$h_{f,f}^{-1}{\big{|}}_{(-1,1)}\in W^{\omega}_{+}((-1,1))$. Then the following
statements hold:

  1. (i)

$\big{(}h_{f,f}^{-1}{\big{|}}_{(-1,1)}\big{)}_{\text{abs}}(y)\geq\frac{\pi}{2}\,\frac{1}{\|f\|_{\infty}^{2}}\,y\text{
for all }y\in[0,1]\,.$

  2. (ii)

$\displaystyle
K_{G}^{\mathbb{R}}\leq\frac{\|f\|_{\infty}^{2}}{h_{f,f}^{\text{hyp}}(1)}\,.$

  3. (iii)

Let $1\leq c_{\ast}<K_{G}^{\mathbb{R}}$. If $\|f\|_{\infty}=1$, then
$0<h_{f,f}^{\text{hyp}}(c_{\ast})<1$ and there is exactly one number
$\gamma^{\ast}(f)\in(h_{f,f}^{\text{hyp}}(c_{\ast}),1]$, such that

$\displaystyle
K_{G}^{\mathbb{R}}=\big{(}h_{f,f}^{-1}{\big{|}}_{(-1,1)}\big{)}_{\text{abs}}(\gamma^{\ast}(f))\leq\min\Big{\\{}\frac{1}{h_{f,f}^{\text{hyp}}(1)},\,\big{(}h_{f,f}^{-1}{\big{|}}_{(-1,1)}\big{)}_{\text{abs}}(1)\Big{\\}}\,.$
(6.201)

###### Proof.

Since $f\in L^{\infty}$, it follows that $f\in L^{2}(\gamma_{k})$ and
$1=\|f\|_{\gamma_{k}}\leq\|f\|_{\infty}$.
(i) and (ii) The claimed two inequalities directly follow from (5.123),
(6.158), Lemma 6.11, Theorem 6.40, and (6.200).
(iii) Since $1\leq
c_{\ast}<K_{G}^{\mathbb{R}}\leq\big{(}h_{f,f}^{-1}{\big{|}}_{(-1,1)}\big{)}_{\text{abs}}(1)$
(due to Theorem 6.40), it follows from Lemma 6.11 that
$0<h_{f,f}^{\text{hyp}}(1)\leq
h_{f,f}^{\text{hyp}}(c_{\ast})<\gamma^{\ast}(f):=h_{f,f}^{\text{hyp}}(K_{G}^{\mathbb{R}})\leq
1$. (iii) now follows from (ii). ∎

If we only assume that $f\in L^{\infty}(\mathbb{R}^{k})\setminus\\{0\\}$, then
an application of Theorem 6.45 to $\frac{1}{\|f\|_{\gamma_{k}}}f\in
S_{L^{2}(\gamma_{k})}$ directly leads to

######  Corollary 6.46.

Let $k\in\mathbb{N}$ and $f\in L^{\infty}(\mathbb{R}^{k})\setminus\\{0\\}$.
Then $0<r\equiv r_{k}(f):=\|f\|_{\gamma_{k}}^{2}<\infty$ and
$\|f\|_{\infty}\geq\sqrt{r}$. Assume that $f$ is odd and
$h_{f,f}^{-1}{\big{|}}_{(-r,r)}\in W^{\omega}_{+}((-r,r))$. Then
$\big{(}h_{f,f}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(r)>1$, and the
following statements hold:

  1. (i)

$\big{(}h_{f,f}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(y)\geq\frac{\pi}{2}\,\frac{1}{\|f\|_{\infty}^{2}}\,y\,\text{
for all }\,y\in[0,r]\,.$

  2. (ii)

$\displaystyle
r\,K_{G}^{\mathbb{R}}\leq\frac{\|f\|_{\infty}^{2}}{h_{\frac{f}{\sqrt{r}},\frac{f}{\sqrt{r}}}^{\text{hyp}}(1)}\,.$
(6.202)

  3. (iii)

Let $1\leq c_{\ast}<K_{G}^{\mathbb{R}}$. If $\|f\|_{\infty}=\sqrt{r}$, then
$0<h_{\frac{f}{\sqrt{r}},\frac{f}{\sqrt{r}}}^{\text{hyp}}(c_{\ast})<1$ and
there is exactly one number number
$\gamma^{\ast}(f)\in(h_{\frac{f}{\sqrt{r}},\frac{f}{\sqrt{r}}}^{\text{hyp}}(c_{\ast}),1]$,
such that

$\displaystyle
K_{G}^{\mathbb{R}}=\big{(}h_{f,f}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(r\,\gamma^{\ast}(f))\leq\min\Big{\\{}\frac{1}{h_{\frac{f}{\sqrt{r}},\frac{f}{\sqrt{r}}}^{\text{hyp}}(1)},\,\big{(}h_{f,f}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(r)\Big{\\}}\,.$

The proof of Theorem 6.45-(i) shows us that here we may circumvent the rather
strong assumption of $h_{f,f}^{-1}{\big{|}}_{(-r,r)}$ being completely real
analytic on $(-r,r)$ at $0$; at least in the following sense (cf. Example 6.51
as application for this):

######  Proposition 6.47.

Let $k\in\mathbb{N}$ and $f\in L^{\infty}(\mathbb{R}^{k})\setminus\\{0\\}$.
Then $0<r\equiv r_{k}:=\|f\|_{\gamma_{k}}^{2}<\infty$ and
$\|f\|_{\infty}\geq\sqrt{r}$. Assume that $f$ is odd, $h_{f,f}^{\prime}(0)>0$
and

$\displaystyle\sum_{n=0}^{\infty}\frac{|(h_{f,f}^{-1})^{(2n+1)}(0)|}{(2n+1)!}\,(c^{\ast})^{2n+1}=1\,,$
(6.203)

for some $c^{\ast}\in(0,r]$. Then
$h_{f,f}^{-1}{\big{|}}_{(-c^{\ast},c^{\ast})}\in
W^{\omega}_{+}((-c^{\ast},c^{\ast}))$, and the following statements hold:

  1. (i)

$\big{(}h_{f,f}^{-1}{\big{|}}_{(-c^{\ast},c^{\ast})}\big{)}_{\text{abs}}(y)\geq\frac{\pi}{2}\,\frac{1}{\|f\|_{\infty}^{2}}\,y\,\text{
for all }\,y\in[0,c^{\ast}]\,\text{ and
}\,\big{(}h_{f,f}^{-1}{\big{|}}_{(-c^{\ast},c^{\ast})}\big{)}_{\text{abs}}(c^{\ast})=1\,.$

  2. (ii)

$\displaystyle h_{f,f}^{-1}[c^{\ast}\,S]\in{\mathcal{Q}}_{m,n}\,\text{ for all
}\,S\in{\mathcal{Q}}_{m,n}\,.$ (6.204)

  3. (iii)

$\displaystyle K_{G}^{\mathbb{R}}\leq\frac{\|f\|_{\infty}^{2}}{c^{\ast}}\,.$

###### Proof.

(i) The assumption clearly implies that
$h_{f,f}^{-1}{\big{|}}_{(-c^{\ast},c^{\ast})}\in
W^{\omega}_{+}((-c^{\ast},c^{\ast}))$ and
$\big{(}h_{f,f}^{-1}{\big{|}}_{(-c^{\ast},c^{\ast})}\big{)}_{\text{abs}}(c^{\ast})=1$.
(ii) and (iii) Observe that in particular the restriction
$h_{f,f}^{-1}{\big{|}}_{[-c^{\ast},c^{\ast}]}:[-c^{\ast},c^{\ast}]\longrightarrow[-1,1]$
is continuous. Consequently, if $S\in\mathcal{Q}_{mn}$, then
$S_{0}:=h_{f,f}^{-1}[c^{\ast}\,S]\in\mathcal{Q}_{mn}$ (due to (6.188)), and
(ii) follows. Let $m,n\in\mathbb{N}$, $A\in\mathbb{M}_{m,n}(\mathbb{R})$ and
$S\in\mathcal{Q}_{m,n}$ be arbitrarily given. Since
$S_{0}\in\mathcal{Q}_{mn}$, we therefore obtain

$\displaystyle|{\text{tr}}(A^{\top}S)|$
$\displaystyle=\frac{1}{c^{\ast}}\,|{\text{tr}}(A^{\top}h_{f,f}[S_{0}])|\stackrel{{\scriptstyle\eqref{eq:property_of_quantum_corr_matrices}}}{{\leq}}\frac{1}{c^{\ast}}\,\|A\|_{\infty,1}\,,$

and (iii) follows as well. ∎

######  Example 6.48 (Grothendieck and Krivine).

Once again, we consider the CCP function $\psi:=\frac{2}{\pi}\arcsin$. Recall
that $\psi=h_{f,f}$, where $f:=\text{sign}\in S_{L^{\infty}(\mathbb{R})}\cap
S_{L^{2}(\gamma_{k})}$. Due to (6.162), it follows that

$\psi^{\text{hyp}}(1)=\frac{2}{\pi}\,\sinh^{-1}(1)=\frac{2}{\pi}\,\ln(1+\sqrt{2}).$

Hence,

$K_{G}^{\mathbb{R}}\stackrel{{\scriptstyle\eqref{eq:upper_bound_real_odd_unit_sphere_case}}}{{\leq}}\min\Big{\\{}\frac{1}{\psi^{\text{hyp}}(1)},\,\big{(}\psi^{-1}{\big{|}}_{(-1,1)}\big{)}_{\text{abs}}(1)\Big{\\}}\leq\frac{\pi}{2\ln(1+\sqrt{2})}(\approx
1.78221)\leq\sinh(\frac{\pi}{2})(\approx 2.30129)$

precisely reflects Krivine’s upper bound of $K_{G}^{\mathbb{R}}$ as well as
Grothendieck’s (larger) upper bound of $K_{G}^{\mathbb{R}}$ !

######  Example 6.49.

Consider the CCP function $\kappa:=\sqrt{3}(2\phi-1)$ (cf. Proposition 6.23).
Due to Theorem 6.45, (i), respectively (6.202), we obtain the following
(weaker) estimation:

$K_{G}^{\mathbb{R}}\leq\frac{\pi}{6\,\ln(\tfrac{1}{2}(1+\sqrt{5}))}\cdot\|\kappa\|_{\infty}^{2}\leq\frac{\pi}{2\,\ln(\tfrac{1}{2}(1+\sqrt{5}))}(\approx
3.26425)\,.$

We highly recommend the reader to check whether this estimation can be
improved, if more generally the function $\kappa_{\alpha}:=\alpha(2\phi-1)$ is
considered, where $0<\alpha<\sqrt{3}$ is given (instead of the CCP function
$\kappa=\kappa_{\sqrt{3}}$)! Observe also that
$\|\kappa\|_{\infty}=\sqrt{3}\not=1=\|\kappa\|_{\gamma_{k}}$.

######  Example 6.50.

Fix $k\in\mathbb{N}_{3}$ and consider the function $\psi:=h_{f_{k},f_{k}}$,
introduced in Proposition 6.16. Assume that $\psi^{-1}\in
W^{\omega}_{+}((-1,1))$. $\psi$ then satisfies all assumptions, listed in
Theorem 6.45, and it follows that
$0<c^{\ast}_{k}:=h_{f_{k},f_{k}}^{\text{hyp}}(1)<1$ satisfies

$\displaystyle K_{G}^{\mathbb{R}}\leq\frac{k}{c^{\ast}_{k}}$ (6.205)

(since $\|f_{k}\|_{\infty}^{2}=k$). However, observe that the sequence
$\big{(}\frac{k}{c^{\ast}_{k}}\big{)}_{k\in\mathbb{N}}$ is not bounded and
hence cannot converge (since $0<c_{k}^{\ast}\leq 1$). Moreover, in contrast to
the previous two examples, we do not know whether also
$\big{(}h_{f_{k},f_{k}}^{-1}{\big{|}}_{(-1,1)}\big{)}_{\text{abs}}$ can be
represented in a closed analytical form; one of the major open problems in our
search for the smallest upper bound of $K_{G}^{\mathbb{R}}$ (cf. Subsection
9.1 and Example 7.15, where the latter includes the approximation of the
constant $c_{k}^{\ast}\approx 0.71200$ if $k=2$). A straightforward, yet a bit
laborious calculation with fractions, based on the table (9.249) (similarly to
the special case $k=2$, treated in Example 7.15), yields that the Maclaurin
series of
$h_{f_{k},f_{k}}^{-1}(s)\equiv\sum_{\nu=0}^{\infty}\beta_{2\nu+1}(k)\,s^{2\nu+1}$
can e.g. be approximated by the Taylor polynomial of degree 7 as:

$\displaystyle\begin{split}h_{f_{k},f_{k}}^{-1}(s)&=\beta_{1}(k)s+\beta_{3}(k)s^{3}+\beta_{5}(k)s^{5}+\beta_{7}(k)s^{7}+o(|s|^{7})\\\
&=\frac{1}{c_{k}^{2}}\,s+\big{(}-\frac{1}{c_{k}^{6}}\frac{1}{2(k+2)}\,s^{2}\big{)}\big{(}s+\frac{3}{4c_{k}^{4}}\,\frac{k-2}{(k+2)(k+4)}\,s^{3}\\\
&+\frac{3}{8c_{k}^{8}}\,\frac{9(k+2)^{2}-48(k+2)+64}{(k+2)^{2}\,(k+4)\,(k+6)}\,s^{5}\big{)}+o(|s|^{7}).\end{split}$
(6.206)

Observe also that
$\beta_{5}(k):=-\frac{3}{8c_{k}^{10}}\,\frac{k-2}{(k+2)^{2}\,(k+4)}=0$ if and
only if $k=2$ and that $\beta_{5}(k)\leq 0$ if and only if $k\geq 2$.
Similarly, since the single (local) minimum of the function $\mathbb{R}\ni
x\mapsto g(x):=9x^{2}-48x+64$ is attained at $x_{\ast}:=\frac{8}{3}$ and
$g(x_{\ast})=0$, it follows that $\beta_{7}(k)\leq 0$.

######  Example 6.51 (Shortening of the proof of [21, Theorem 1.1]).

Our approach also allows to recover the strongest result to date, namely that
$K_{G}^{\mathbb{R}}<\frac{\pi}{2\ln(1+\sqrt{2})}$; at least partially. What is
certain, however, is that our approach also allows a non-negligible shortening
of the proof of the key result [21, Theorem 1.1]. To this end, firstly observe
that a simple change of variables reveals that for any $f,g\in
L^{2}(\gamma_{k})$, the corresponding generalised function
$H_{f\circ\sqrt{2},g\circ\sqrt{2}}$, listed in [21, Definition 2.1], satisfies

$\displaystyle H_{f\circ\sqrt{2},g\circ\sqrt{2}}=h_{f,g}\big{|}_{(-1,1)}$
(6.207)

(due to (6.147)). So, $H_{f\circ\sqrt{2},g\circ\sqrt{2}}$ is well-defined on
$(-1,1)$. The Grothendieck function $\frac{2}{\pi}\arcsin$ is then generalised
in [21] to the complex-valued function

$\displaystyle F_{p,\eta}:=(1-p)H_{0}+pH_{\eta}\,,$ (6.208)

where $0\leq p\leq 1$, $0\leq\eta<1$ and
$H_{\eta}:\mathbb{S}\longrightarrow\mathbb{C}$ is defined as in [21, (41)],
where $\mathbb{S}:=\\{z\in\mathbb{C}:|\operatorname{Re}(z)|<1\\}$. Independent
of any complex analysis a priori, the construction of $H_{\eta}$, together
with (6.207) implies that

$H_{\eta}\big{|}_{\mathbb{S}\cap\mathbb{R}}=H_{\eta}\big{|}_{(-1,1)}=H_{g_{\eta}\circ\sqrt{2},g_{\eta}\circ\sqrt{2}}\stackrel{{\scriptstyle\eqref{eq:Krivine_RS_vs_h_f_g}}}{{=}}h_{g_{\eta},g_{\eta}}\big{|}_{(-1,1)}\,.$

Here, the odd function $g_{\eta}:\mathbb{R}^{2}\longrightarrow\mathbb{R}$ is
defined as $g_{\eta}(x):=\sqrt{\frac{\pi}{2}}\,f_{\eta}(\frac{1}{\sqrt{2}}x)$,
where the 2-dimensional sign concept
$f_{\eta}:\mathbb{R}^{2}\longrightarrow\\{-1,1\\}$ satisfies [21, (40)].
Consequently, the function $\frac{2}{\pi}\,H_{\eta}\big{|}_{(-1,1)}$ actually
is a restriction of the well-defined odd - and hence invertible - CCP function
$\frac{2}{\pi}\,h_{g_{\eta},g_{\eta}}=h_{\sqrt{2/\pi}\,g_{\eta},\sqrt{2/\pi}\,g_{\eta}}$
to the open interval $(-1,1)$ (due to Theorem 6.33).

Let $\rho\in(-1,1)$ and ${\text{vec}}(\textbf{X},\textbf{Y})\sim
N_{4}(0,\Sigma_{4}(\rho))$. Since
$g_{0}(x)=\sqrt{\frac{\pi}{2}}\,{\text{sign}}(x_{2})$ for all
$x=(x_{1},x_{2})^{\top}\in\mathbb{R}^{2}$, we may apply Proposition 2.13 to
the partitioned Gaussian random vector ${\text{vec}}(\textbf{X},\textbf{Y})$,
and it follows that

$H_{0}(\rho)=h_{g_{0},g_{0}}(\rho)=\frac{\pi}{2}\,\mathbb{E}_{\tiny{\mathbb{P}_{{\text{vec}}(\textbf{X},\textbf{Y})}}}[\text{sign}(X_{2})\,\text{sign}(Y_{2})]=\arcsin(\rho),$

which already gives a one-line proof of [21, Lemma 4.3]. In particular, for
any $p\in[0,1]$,

$\displaystyle\frac{2}{\pi}\,F_{p,\eta}\big{|}_{(-1,1)}=\psi_{p,\eta}\big{|}_{(-1,1)}$
(6.209)

emerges as a restriction of the odd and hence strictly increasing,
homeomorphic CCP function
$\psi_{p,\eta}:=(1-p)\frac{2}{\pi}\arcsin+p\,\frac{2}{\pi}\,h_{g_{\eta},g_{\eta}}$
on $(-1,1)$ (due to Theorem 5.17 and Theorem 6.8-(iii)). Observe that
$\psi_{p,\eta}^{\prime}(0)\geq(1-p)\frac{2}{\pi}$, implying that
$\psi_{p,\eta}^{-1}\big{|}_{(-1,1)}\in C^{\omega}((-1,1))$ if $p<1$ (due to
Theorem 6.8-(iv)). Because of the non-trivial result [21, Theorem 5.1]
(including its technically demanding proof) it follows the existence of
$(p_{0},\eta_{0})\in(0,1)\times(0,1)$ and
$c^{\ast}\in(\frac{2}{\pi}\ln(1+\sqrt{2}),\frac{9}{5\pi})$, such that

$\sum_{n=0}^{\infty}\frac{|(\psi_{p_{0},\eta_{0}}^{-1})^{(2n+1)}(0)|}{(2n+1)!}\,(c^{\ast})^{2n+1}=1\,.$

(Given the outcome of [21, Theorem 5.1], we only have to set
$c^{\ast}:=\frac{2\gamma}{\pi}$, where $\ln(1+\sqrt{2})<\gamma<\frac{9}{10}$
satisfies [21, (63)].) Hence,
$\psi_{p_{0},\eta_{0}}^{-1}\big{|}_{(-c^{\ast},c^{\ast})}\in
W^{\omega}_{+}((-c^{\ast},c^{\ast}))$. Since in particular
$\psi_{p_{0},\eta_{0}}^{-1}\big{|}_{[-c^{\ast},c^{\ast}]}$ is continuous, we
may apply Proposition 6.47. Consequently, if $S\in\mathcal{Q}_{mn}$, then
$S_{0}:=\psi_{p_{0},\eta_{0}}^{-1}[c^{\ast}\,S]\in\mathcal{Q}_{mn}$ (due to
(6.204)). Observe that for any $0\leq\eta<1$ the CCP function
$\frac{2}{\pi}\,h_{g_{\eta},g_{\eta}}=h_{\sqrt{2/\pi}\,g_{\eta},\sqrt{2/\pi}\,g_{\eta}}$
actually originates from the bounded function
$\sqrt{\frac{2}{\pi}}\,g_{\eta}\in S_{L^{\infty}}\cap S_{L^{2}(\gamma_{2})}$,
such as $\frac{2}{\pi}\arcsin=h_{\text{sign},\text{sign}}$. Let
$S\in\mathcal{Q}_{m,n}$ be arbitrarily given. Since
$S_{0}\in\mathcal{Q}_{mn}$, we therefore obtain

$\displaystyle|{\text{tr}}(A^{\top}S)|$
$\displaystyle=\frac{1}{c^{\ast}}\,|{\text{tr}}(A^{\top}\psi_{p_{0},\eta_{0}}[S_{0}])|=\frac{1}{c^{\ast}}\,\big{|}{\text{tr}}(A^{\top}\big{(}(1-p)\frac{2}{\pi}\arcsin[S_{0}]+p\,\frac{2}{\pi}\,h_{g_{\eta},g_{\eta}}[S_{0}]\big{)}\big{|}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:property_of_quantum_corr_matrices}}}{{\leq}}\frac{1}{c^{\ast}}\,((1-p)\|A\|_{\infty,1}+p\,\|A\|_{\infty,1})=\frac{1}{c^{\ast}}\,\|A\|_{\infty,1}\,,$

whence

$K_{G}^{\mathbb{R}}\leq\frac{1}{c^{\ast}}<\frac{\pi}{2\,\ln(1+\sqrt{2})}\,.$

In summary, given the - crucial - result [21, Theorem 5.1], our general
framework can be applied here as well, leading to a shortened proof of [21,
Theorem 1.1].

Despite the quite remarkable outcome of Theorem 6.45, we should observe that
its practical implementation seems to be quite difficult (at least without
sufficiently large computer power). Primarily, as we already have seen, this
is due to the following facts:

  1. (i)

Either we have to know a closed form representation (or at least a “close”
approximation of the Maclaurin series) of $h_{f,f}$, $h_{f,f}^{-1}$ and
$\big{(}h_{f,f}^{-1}\big{)}_{\text{abs}}$ if $f\in L^{\infty}(\mathbb{R}^{k})$
is given (such as is the case for $k=1$ and $f:=\text{sign}\in
L^{\infty}(\mathbb{R}^{1})\cap S_{L^{2}(\gamma_{1})}$), or we have to check,
whether $h=h_{f,f}$ “originates” from some $f\in
L^{\infty}(\mathbb{R}^{k})\cap S_{L^{2}(\gamma_{k})}$, if the functions $h$,
$h^{-1}$ and $\big{(}h^{-1}\big{)}_{\text{abs}}$ are known to us.

  2. (ii)

However, already in the one-dimensional case (i.e., for $k=1$) that search
requires rather complex calculation techniques, respectively some very helpful
knowledge about Hermite polynomials. Moreover, if $k$ increases, we are
confronted with a “curse of combinatoric dimensionality”, since for any
$\nu\in\mathbb{N}_{0}$ and $n\in\mathbb{N}$ it can be easily shown by
induction on $k$ that the set $C(\nu,k):=\\{n\in\mathbb{N}_{0}^{k}:|n|=\nu\\}$
which determines the structure of $h_{f,f}$ (cf. Theorem 6.5) actually
consists of $\binom{\nu+k-1}{k-1}=\frac{(\nu+k-1)!}{\nu!\,(k-1)!}$ elements.
In particular, already $C(\nu,2)$ consists of $\nu+1$ elements
($\nu\in\mathbb{N}_{0}$). For example, to determine $C(2,11)$ explicitly, we
would have to know all of its $66$ elements!

We will recognise how deep actually we are confronted with a “curse of
combinatoric dimensionality” if only a Maclaurin series representation of the
function $h_{f,f}$ is given to us (cf. Subsection 9.1).

##  7 The complex case: towards extending Haagerup’s approach

###  7.1 Multivariate complex CCP functions and their relation to the real
case

Next, we are going to transfer the main results in the previous section from
the real field $\mathbb{R}$ to the complex field $\mathbb{C}$. In order to
achieve this, we have to implement non-trivial structural properties of the
class of complex Hermite polynomials (cf. Theorem 7.11 and Theorem 7.12). The
complex versions of Theorem 6.5 and Theorem 6.8 also allow a generalisation of
the Haagerup equality by transition from the $\Sigma_{2}(\zeta)$-correlated
couple of two complex one-dimensional signum functions
${\text{sign}}:\mathbb{C}\longrightarrow\mathbb{T}$ to a
$\Sigma_{2k}(\zeta)$-correlated couple of two (possibly different) arbitrary
square-integrable functions $b,c:\mathbb{C}^{k}\longrightarrow\mathbb{C}$,
where $k\in\mathbb{N}$ can be arbitrarily large (cf. Corollary 7.4 and Example
7.15). By definition (cf. [56, Lemma 3.2. and Proof of Theorem 3.1]), the
complex sign-function is given as

${\text{sign}}(z):=\begin{cases}\frac{z}{|z|}&\text{if
}z\in\mathbb{C}^{\ast}\\\ 0&\text{if }z=0\end{cases}\,.$

We introduce the following helpful symbolic constructions and shortcuts. Fix
$k,l\in\mathbb{N}$. Let $b:\mathbb{C}^{k}\longrightarrow\mathbb{C}$ and
$c:\mathbb{C}^{l}\longrightarrow\mathbb{C}$ be two functions. Put

$b{}_{k}\\!\otimes_{l}c:=(b\circ P_{k})(c\circ Q_{l}),$

where

$P_{k}:=\begin{pmatrix}1&0&\ldots&0&0&0&\ldots&0\\\
0&1&\ldots&0&0&0&\ldots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\ldots&1&0&0&\ldots&0\end{pmatrix}=(I_{k}\,\brokenvert\,0)\in\mathbb{M}_{k,k+l}(\mathbb{C})$

and

$Q_{l}:=\begin{pmatrix}0&0&\ldots&0&1&0&\ldots&0\\\
0&0&\ldots&0&0&1&\ldots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\ldots&0&0&0&\ldots&1\end{pmatrix}=(0\,\brokenvert\,I_{l})\in\mathbb{M}_{l,k+l}(\mathbb{C}),$

implying that

$\displaystyle
b\,{}_{k}\\!\otimes_{l}c({\text{vec}}(z\,\brokenvert\,w))=b(z)c(w)\text{ for
all }(z,w)\in\mathbb{C}^{k}\times\mathbb{C}^{l}\,.$ (7.210)

Given the construction of $b\,{}_{k}\\!\otimes_{l}c$ we may unambiguously
shorten it simply to $b\otimes c$ (and suppress the listing of the dimensions
of the domains of definition of $b$, respectively $c$). Let
$d:\mathbb{C}^{k}\longrightarrow\mathbb{C}$ be a given function. Recall the
induced functions $r(d):\mathbb{R}^{2k}\longrightarrow\mathbb{R}$ and
$s(d):\mathbb{R}^{2k}\longrightarrow\mathbb{R}$, defined as
$r(d):=\operatorname{Re}(d)\circ\frac{1}{\sqrt{2}}\,J_{2}^{-1}$ and
$s(d):=\operatorname{Im}(d)\circ\frac{1}{\sqrt{2}}\,J_{2}^{-1}$ (cf. (2.24)).
In order to facilitate reading, we put
$d_{\alpha}(z):=d({\text{sign}}(\alpha)z)$, where $\alpha\in\mathbb{C}$ and
$z\in\mathbb{C}^{k}$ (implying that $d_{0}=d(0)$). Consequently, the
construction of $b\otimes\overline{c}$ implies that

$\displaystyle
r(b\otimes\overline{c})({\text{vec}}({\text{vec}}(x_{1},x_{2}),{\text{vec}}(y_{1},y_{2})))$
$\displaystyle=r(b)({\text{vec}}(x_{1},y_{1}))r(c)({\text{vec}}(x_{2},y_{2}))$
$\displaystyle+s(b)({\text{vec}}(x_{1},y_{1}))s(c)({\text{vec}}(x_{2},y_{2}))$

and

$\displaystyle
s(b\otimes\overline{c})({\text{vec}}({\text{vec}}(x_{1},x_{2}),{\text{vec}}(y_{1},y_{2})))$
$\displaystyle=s(b)({\text{vec}}(x_{1},y_{1}))r(c)({\text{vec}}(x_{2},y_{2}))$
$\displaystyle-r(b)({\text{vec}}(x_{1},y_{1}))s(c)({\text{vec}}(x_{2},y_{2}))$

for all $x_{1},x_{2},y_{1},y_{2}\in\mathbb{R}^{k}$. In other words:

$\displaystyle r(b\otimes\overline{c})=(r(b)\otimes r(c))\circ G+(s(b)\otimes
s(c))\circ G$ (7.211)

on $\mathbb{R}^{4k}$, and

$\displaystyle s(b\otimes\overline{c})=(s(b)\otimes r(c))\circ G-(r(b)\otimes
s(c))\circ G\,,$ (7.212)

on $\mathbb{R}^{4k}$, where again $G=G^{\top}=G^{-1}\in O(4n)$ is the matrix,
introduced in (1.9). Finally, $d$ is odd (respectively, even) if and only if
both, $r(d)$ and $s(d)=r(-i\,d)$ are odd (respectively, even).

######  Lemma 7.1.

Let $k\in\mathbb{N}$, $\rho\in[-1,1]$, $\alpha\in\mathbb{C}$,
$\theta\in\mathbb{R}$, $c,d:\mathbb{C}^{k}\longrightarrow\mathbb{C}$ and
${\text{vec}}(\textbf{X},\textbf{Y})\sim N_{4k}(0,\Sigma_{4k}(\rho))$. Then

  1. (i)

$r(d_{\alpha})=r(d)\circ R_{2}({\text{sign}}(\alpha)I_{k})$ and
$r(d_{\theta})=r(d)\circ{\text{sign}}(\theta)I_{2k}$.

  2. (ii)

If $\alpha\not=0$, then $r(d)\in L^{2}(\mathbb{R}^{2k},\gamma_{2k})$ if and
only if $r(d_{\alpha})\in L^{2}(\mathbb{R}^{2k},\gamma_{2k})$. In this case,
the norms coincide: $\|r(d)\|_{\gamma_{2k}}=\|r(d_{\alpha})\|_{\gamma_{2k}}$.

  3. (iii)

If $\alpha\not=0$,
$h_{r(d),r(d)}(\rho)=\mathbb{E}[r(d)(\textbf{X})r(d)(\textbf{Y})]=\mathbb{E}[r(d_{\alpha})(\textbf{X})r(d_{\alpha})(\textbf{Y})]=h_{r(d_{\alpha}),r(d_{\alpha})}(\rho)$.

  4. (iv)

$h_{r(c_{r}),r(d)}(|r|)=h_{r(c),r(d)}(r)\text{ for all }r\in[-1,1]$.

###### Proof.

(i) Put $\kappa:={\text{sign}}(\alpha)$. Fix $x\in\mathbb{R}^{2k}$. Since
$J_{2}^{-1}\circ R_{2}(\kappa\,I_{k})=(\kappa\,I_{k})\circ
J_{2}^{-1}=\kappa\,J_{2}^{-1}$ (cf. Figure 1), it follows that
$d_{\alpha}\big{(}\frac{1}{\sqrt{2}}\,J_{2}^{-1}(x)\big{)}=d\big{(}\frac{1}{\sqrt{2}}J_{2}^{-1}(R_{2}(\kappa\,I_{k})x)\big{)}$,
whence $r(d_{\alpha})=r(d)\circ R_{2}(\kappa I_{k})$.
(ii) Let $\alpha\not=0$. Since $\det(R_{2}(\kappa I_{k}))=1$ (due to (2.22))
and $\|R_{2}(\kappa I_{k})a\|^{2}=\langle
a,R_{2}(\overline{\kappa}I_{k})R_{2}(\kappa
I_{k})a\rangle_{\mathbb{R}_{2}^{2k}}=\|a\|^{2}$ for all $a\in\mathbb{R}^{2k}$
(due to (1.14)), an application of the change-of-variables formula implies
that $r(d_{\alpha})\in L^{2}(\mathbb{R}^{2k},\gamma_{2k})$ if and only if
$r(d)\in L^{2}(\mathbb{R}^{2k},\gamma_{2k})$ and
${\|r(d_{\alpha})\|}_{2}={\|r(d)\|}_{2}$.
(iii) Let $\alpha\not=0$. Put $\kappa:={\text{sign}}(\alpha)$. Then
$A_{\kappa}:=R_{2}(\kappa I_{k})\not=0$. Put
${\textbf{X}}_{\kappa}:=A_{\kappa}\textbf{X}$ and
${\textbf{Y}}_{\kappa}:=A_{\kappa}\textbf{Y}$. Since

$\begin{pmatrix}A_{\kappa}&0\\\
0&\,A_{\kappa}\end{pmatrix}\begin{pmatrix}I_{2k}&\rho\,I_{2k}\\\
\rho\,I_{2k}&\,I_{2k}\end{pmatrix}\begin{pmatrix}A_{\overline{\kappa}}&0\\\
0&A_{\overline{\kappa}}\end{pmatrix}=\begin{pmatrix}I_{2k}&\rho\,I_{2k}\\\
\rho\,I_{2k}&\,I_{2k}\end{pmatrix}=\Sigma_{4k}(\rho),$

it follows that

${\text{vec}}({\textbf{X}}_{\kappa},{\textbf{Y}}_{\kappa})=\begin{pmatrix}A_{\kappa}&0\\\
0&\,A_{\kappa}\end{pmatrix}{\text{vec}}(\textbf{X},\textbf{Y})\sim
N_{4k}(0,\Sigma_{4k}(\rho)).$

Therefore, ${\text{vec}}(\textbf{X},\textbf{Y})\stackrel{{\scriptstyle
d}}{{=}}{\text{vec}}({\textbf{X}}_{\kappa},{\textbf{Y}}_{\kappa})$, whence
$\mathbb{E}[r(d_{\alpha})(\textbf{X})r(d_{\alpha})(\textbf{Y})]\stackrel{{\scriptstyle(i)}}{{=}}\mathbb{E}[r(d)({\textbf{X}}_{\kappa})r(d)({\textbf{Y}}_{\kappa})]=\mathbb{E}[r(d)(\textbf{X})r(d)(\textbf{Y})]$
which proves (iii).
(iv) If $r=0$, the claim immediately follows from the resulting independence
of the standard Gaussian random vectors X and Y. So, let $r\not=0$. Let
${\text{vec}}(\textbf{X}_{1},\textbf{X}_{2})\sim N_{4k}(0,\Sigma_{4k}(|r|))$.
Then ${\text{vec}}({\text{sign}}(r)\textbf{X}_{1},\textbf{X}_{2})\sim
N_{4k}(0,\Sigma_{4k}(r))$. The second equality of statement (i) therefore
implies that

$h_{r(c_{r}),r(d)}(|r|)=\mathbb{E}[r(c_{r})(\textbf{X}_{1})r(d)(\textbf{X}_{2})]=\mathbb{E}[r(c)({\text{sign}}(r)\textbf{X}_{1})r(d)(\textbf{X}_{2})]=h_{r(c),r(d)}(r).$

∎

Fix
${\text{vec}}(\textbf{Z},\textbf{W})\sim{\mathbb{C}}N_{2k}(0,\Sigma_{2k}(\zeta))$,
where $\zeta\in\overline{\mathbb{D}}\setminus\\{0\\}$ and $k\in\mathbb{N}$.
Let $b,c:\mathbb{C}^{k}\longrightarrow\mathbb{C}$, such that
$b\otimes\overline{c}\in
L^{1}(\mathbb{C}^{2k},\mathbb{P}_{{\text{vec}}(\textbf{Z},\textbf{W})})$. We
put

$h^{\mathbb{C}}_{b,c}(\zeta):=h^{\mathbb{C}}(b,c;\zeta):=\mathbb{E}\big{[}b(\textbf{Z})\overline{c(\textbf{W})}\big{]}=\overline{h^{\mathbb{C}}_{c,b}(\overline{\zeta})}.$

As in the real case (see (6.155)), the joint multivariate Gaussian splitting
property (3.92) of inner products of vectors on the unit sphere and Lemma
2.14-(ii) imply the important observation that for any separable
$\mathbb{C}$-Hilbert space $H$, for any $m,n\in\mathbb{N}$, and for any
$(u,v)\in S_{H}^{m}\times S_{H}^{n}$, we have

$\displaystyle
h^{\mathbb{C}}_{b,c}[\Gamma_{H}(u,v)]=\mathbb{E}\big{[}\overline{\textbf{R}_{c}}\,{\textbf{S}_{b}}^{\top}\big{]}=\mathbb{E}[\Gamma_{\mathbb{C}}(\textbf{R}_{c},\textbf{S}_{b})],$
(7.213)

where the $\mathbb{C}^{m}$-valued random vector $\textbf{R}_{c}$ and the
$\mathbb{C}^{n}$-valued random vector $\textbf{S}_{b}$ are defined as
$(\textbf{R}_{c})_{i}:=c(\textbf{Z}_{u_{i}})$ and
$(\textbf{S}_{b})_{j}:=b(\textbf{Z}_{v_{j}})$, respectively
($(i,j)\in[m]\times[n]$). Fix $f,g\in L^{2}(\mathbb{R}^{k},\gamma_{k})$ and
put

$H_{f,g}:=h_{f,f}+h_{g,g}.$

(6.147) implies a concrete integral representation of the function
$H_{f,g}{\big{|}}_{(-1,1)}$, which particularly plays an important role in the
complex case (cf. Corollary 7.4):

$\displaystyle
H_{f,g}(\rho)=\frac{1}{(2\pi)^{k}(1-\rho^{2})^{k/2}}\,\int_{\mathbb{R}^{k}}\int_{\mathbb{R}^{k}}\big{\langle}\binom{f(x)}{g(x)},\binom{f(y)}{g(y)}\big{\rangle}_{\mathbb{R}_{2}^{2}}\exp\big{(}-\frac{\|x\|^{2}+\|y\|^{2}-2\rho\langle
x,y\rangle}{2(1-\rho^{2})}\big{)}\textup{d}^{k}x\,\textup{d}^{k}y$ (7.214)

for all $\rho\in(-1,1)$. Theorem 6.5 further implies that
$H_{f,g}(\rho)=\sum_{\nu=0}^{\infty}a_{\nu}\,\rho^{\nu}$ for all
$\rho\in[-1,1]$, where $a_{\nu}:=p_{\nu}(f,f)+p_{\nu}(g,g)\geq 0$ for all
$\nu\in\mathbb{N}_{0}$. In particular,
$H_{f,g}(1)=\|f\|^{2}_{\gamma_{2k}}+\|g\|^{2}_{\gamma_{2k}}$, implying that
$H_{f,g}$ is real analytic on $(-1,1)$, continuous on $[-1,1]$ and absolutely
monotonic on $[0,1]$. Moreover, $H_{f,g}$ is bounded, and

$\displaystyle|H_{f,g}(\rho)|\leq\|f\|^{2}_{\gamma_{2k}}+\|g\|^{2}_{\gamma_{2k}}=H_{f,g}(1)$
(7.215)

for all $\rho\in[-1,1]$. Hence, if $f\not=0$ or $g\not=0$, it follows that

$\displaystyle H_{f,g}=H_{f,g}(1)\,\psi_{f,g},$ (7.216)

where the function
$\psi_{f,g}:=\frac{H_{f,g}}{H_{f,g}(1)}:[-1,1]\longrightarrow[-1,1]$ is CCP
(due to Theorem 5.17). In particular, $H_{f,g}\Big{|}_{(-1,1)}\in
W^{\omega}_{+}((-1,1))$. Thus, if $b\in
L^{2}(\mathbb{C}^{k},\gamma_{k}^{\mathbb{C}})$, then (2.27) implies that
$r(b)\in L^{2}(\mathbb{R}^{2k},\gamma_{2k})$ and $s(b)\in
L^{2}(\mathbb{R}^{2k},\gamma_{2k})$, and

$\displaystyle
H_{r(b),s(b)}(1)=\|r(b)\|^{2}_{\gamma_{2k}}+\|s(b)\|^{2}_{\gamma_{2k}}=\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}}.$
(7.217)

If - in addition - $b$ is odd and $\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}}>0$,
then also $r(b)$ and $s(b)$ are odd functions (such as $H_{r(b),s(b)}$),
satisfying $r(b)\not=0$ or $s(b)\not=0$. Hence, we may apply Theorem 6.8 to
the well-defined odd CCP function $\psi_{r(b),s(b)}$, and it follows that
$H_{r(b),s(b)}:[-1,1]\longrightarrow[-\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}},\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}}]$
is a strictly increasing homeomorphism, such as the inverse function
$(H_{r(b),s(b)})^{-1}:[-\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}},\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}}]\longrightarrow[-1,1]$
(due to (7.217) and (7.216)). Similarly, if we - further - assume that
$H_{r(b),s(b)}^{\prime}(0)>0$, Theorem 6.8-(iv) shows that also
$(H_{r(b),s(b)})^{-1}$ is real analytic on
$(-\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}},\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}})$.
So, we could apply Lemma 5.10 to the real analytic function
$(H_{r(b),s(b)})^{-1}{\big{|}}_{(-\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}},\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}})}$
(if the assumptions are given) to check the existence of
$(H_{r(b),s(b)}^{-1})_{\text{abs}}:[-\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2},\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2}]\longrightarrow\mathbb{R}$
then (a crucial assumption in Theorem 7.13). Equipped with these facts, we
arrive at the complex version of Theorem 6.5:

######  Theorem 7.2.

Let $k\in\mathbb{N}$, $\zeta\in\overline{\mathbb{D}}$,
$\textbf{L}\sim{\mathbb{C}}N_{k}(0,I_{k})$ and
${\text{vec}}(\textbf{Z},\textbf{W})\sim{\mathbb{C}}N_{2k}(0,\Sigma_{2k}(\zeta))$.
Let $b\in L^{2}(\mathbb{C}^{k},\gamma_{k}^{\mathbb{C}})$ and $c\in
L^{2}(\mathbb{C}^{k},\gamma_{k}^{\mathbb{C}})$. Then $b\otimes\overline{c}\in
L^{1}(\mathbb{C}^{k}\times\mathbb{C}^{k},\mathbb{P}_{{\text{vec}}(\textbf{Z},\textbf{W})})$.
If $\zeta=0$, then

$\displaystyle\begin{split}\overline{h^{\mathbb{C}}_{c,b}(0)}&=h^{\mathbb{C}}_{b,c}(0)=\mathbb{E}[b(\textbf{L})]\,\mathbb{E}[\overline{c}(\textbf{L})]\\\
&=h_{r(b),r(c)}(0)+h_{s(b),s(c)}(0)+i\,(h_{s(b),r(c)}(0)-h_{r(b),s(c)}(0)).\end{split}$
(7.218)

If $\zeta\not=0$, then

$\displaystyle\overline{h^{\mathbb{C}}_{c,b}(\overline{\zeta})}=h^{\mathbb{C}}_{b,c}(\zeta)=h_{r(b_{\zeta}),r(c)}(|\zeta|)+h_{s(b_{\zeta}),s(c)}(|\zeta|)+i\,(h_{s(b_{\zeta}),r(c)}(|\zeta|)-h_{r(b_{\zeta}),s(c)}(|\zeta|)).$
(7.219)

In particular,

$\displaystyle 0\leq
h^{\mathbb{C}}_{b_{\overline{\zeta}},b}(\zeta)=H_{r(b),s(b)}(|\zeta|)\text{
and }h^{\mathbb{C}}_{b,c}(1)=\langle b,c\rangle_{\gamma_{k}^{\mathbb{C}}}.$
(7.220)

$h^{\mathbb{C}}_{b,c}:\overline{\mathbb{D}}\longrightarrow\mathbb{C}$ is
bounded and satisfies

$\displaystyle|h^{\mathbb{C}}_{b,c}(\zeta)|\leq\|b\|_{\gamma_{k}^{\mathbb{C}}}\,\|c\|_{\gamma_{k}^{\mathbb{C}}}\text{
for all }\zeta\in\overline{\mathbb{D}}.$ (7.221)

###### Proof.

Firstly, let $\zeta=0$, implying that $\Sigma_{2k}(\zeta)=I_{2k}$. Thus, the
partitioned random vector parts $\textbf{Z}\sim{\mathbb{C}}N_{k}(0,I_{k})$ and
$\textbf{W}\sim{\mathbb{C}}N_{k}(0,I_{k})$ of the complex Gaussian random
vector ${\text{vec}}(\textbf{Z},\textbf{W})\sim{\mathbb{C}}N_{2k}(0,I_{2k})$
are independent (cf. [5, Theorem 2.12]). If $\textbf{X}\stackrel{{\scriptstyle
d}}{{=}}\sqrt{2}J_{2}(\textbf{Z})$ and $\textbf{Y}\stackrel{{\scriptstyle
d}}{{=}}\sqrt{2}J_{2}(\textbf{W})$, then $\textbf{X}\sim N_{2k}(0,I_{2k})$ and
$\textbf{Y}\sim N_{2k}(0,I_{2k})$. (2.25) therefore implies that

$\displaystyle\begin{split}\mathbb{E}\big{[}b(\textbf{Z})\overline{c(\textbf{W})}\big{]}&=\mathbb{E}[b(\textbf{Z})]\mathbb{E}[\overline{c(\textbf{W})}]\\\
&=\mathbb{E}[r(b)(\textbf{X})]\,\mathbb{E}[r(c)(\textbf{Y})]+\mathbb{E}[s(b)(\textbf{X})]\,\mathbb{E}[s(c)(\textbf{Y})]\\\
&+i(\mathbb{E}[s(b)(\textbf{X})]\,\mathbb{E}[r(c)(\textbf{Y})]-\mathbb{E}[r(b)(\textbf{X})]\,\mathbb{E}[s(c)(\textbf{Y})])\\\
&=h_{r(b),r(c)}(0)+h_{s(b),s(c)}(0)+i(h_{s(b),r(c)}(0)-h_{r(b),s(c)}(0)),\end{split}$
(7.222)

where the last equality follows from Theorem 6.5-(i). Next, we consider the
remaining case $\zeta\in\overline{\mathbb{D}}\setminus\\{0\\}$. Put
$\sigma:={\text{sign}}(\overline{\zeta})=\frac{\overline{\zeta}}{|\zeta|}$ and
$\tau:={\text{sign}}(\zeta)=\frac{1}{\sigma}$. Lemma 7.1 implies that
$r(b_{\zeta})\in L^{2}(\mathbb{R}^{2k},\gamma_{2k})$ and
${\|r(b_{\zeta})\|}_{2}={\|r(b)\|}_{2}$. Similarly, it follows that
$s(b_{\zeta})\in L^{2}(\mathbb{R}^{2k},\gamma_{2k})$, and
${\|s(b_{\zeta})\|}_{2}={\|s(b)\|}_{2}$. The proof of (7.219) is built on
Lemma 2.14-(v) and Lemma 7.1-(i). To this end, we have to consider the block
matrix

$G:=\begin{pmatrix}I_{k}&0&0&0\\\ 0&0&I_{k}&0\\\ 0&I_{k}&0&0\\\
0&0&0&I_{k}\end{pmatrix}=G^{\top}=G^{-1}\in O(4k)$

again (cf. (1.9)). Lemma 2.14-(v) implies that
${\text{vec}}(\sigma\,\textbf{Z},\textbf{W})\sim{\mathbb{C}}N_{2k}(0,\Sigma_{2k}(|\zeta|))$
and

$\displaystyle\textbf{X}:=\sqrt{2}\,G\,J_{2}({\text{vec}}(\sigma\,\textbf{Z},\textbf{W}))\stackrel{{\scriptstyle\eqref{eq:impact_of_G}}}{{=}}\sqrt{2}\,{\text{vec}}(J_{2}(\sigma\,\textbf{Z}),J_{2}(\textbf{W}))\sim
N_{4k}(0,\Sigma_{4k}(|\zeta|)).$ (7.223)

Thus, Corollary 2.11, together with the equality $G^{2}=I_{4k}$, yields

$\displaystyle\mathbb{E}[b\otimes\overline{c}({\text{vec}}(\textbf{Z},\textbf{W}))]=\mathbb{E}[b_{\zeta}\otimes\overline{c}({\text{vec}}(\sigma\,\textbf{Z},\textbf{W}))]=\mathbb{E}[r(b_{\zeta}\otimes\overline{c})(G\,\textbf{X})]+i\,\mathbb{E}[s(b_{\zeta}\otimes\overline{c})(G\,\textbf{X})].$
(7.224)

Because of (7.211) and (7.212), it follows in particular that

$r(b_{\zeta}\otimes\overline{c})\circ G=r(b_{\zeta})\otimes
r(c)+s(b_{\zeta})\otimes s(c)$

on $\mathbb{R}^{2k}\times\mathbb{R}^{2k}\cong\mathbb{R}^{4k}$, and

$s(b_{\zeta}\otimes\overline{c})\circ G=s(b_{\zeta})\otimes
r(c)-r(b_{\zeta})\otimes s(c)\,,$

on $\mathbb{R}^{2k}\times\mathbb{R}^{2k}\cong\mathbb{R}^{4k}$. Linearity of
the expectation and Theorem 6.5, together with the fact that by construction
$\textbf{X}={\text{vec}}(\textbf{X}_{1},\textbf{X}_{2})$, (with
$\textbf{X}_{1}:=\sqrt{2}\,J_{2}(\sigma\textbf{Z})$ and and
$\textbf{X}_{2}:=\sqrt{2}\,J_{2}(\textbf{W})$) now imply (7.219). Finally, to
achieve the non-trivial boundedness statement (7.221) (which cannot simply be
derived by making a standard use of the triangle inequality), we need to
implement (a complexification of) the Ornstein-Uhlenbeck semigroup,
respectively (6.153). Because of (7.218), the case $\zeta=0$ is trivial,
though. So, let us fix $0\not=\zeta\in\overline{\mathbb{D}}$. Let
$f\in\\{r(b_{\zeta}),s(b_{\zeta})\\}$ and $g\in\\{r(c),s(c)\\}$. Then
$h_{f,g}(|\zeta|)\stackrel{{\scriptstyle\eqref{eq:OU_correl_rep_incl_boundary}}}{{=}}\langle
f,R_{\zeta}g\rangle_{\gamma_{k}}$, where $R_{\zeta}:=T_{\vartheta(|\zeta|)}$.
Consequently, since
$r(R_{\zeta}g\circ\sqrt{2}J_{2})=\operatorname{Re}(R_{\zeta}g\circ\sqrt{2}J_{2})\circ\frac{1}{\sqrt{2}}J_{2}^{-1}=R_{\zeta}g=\operatorname{Im}(R_{\zeta}g\circ\sqrt{2}J_{2})\circ\frac{1}{\sqrt{2}}J_{2}^{-1}=s(R_{\zeta}g\circ\sqrt{2}J_{2})$,
(2.30) leads to

$\displaystyle h_{b,c}^{\mathbb{C}}(\zeta)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:CCP_complex_correl_case}}}{{=}}h_{r(b_{\zeta}),r(c)}(|\zeta|)+h_{s(b_{\zeta}),s(c)}(|\zeta|)+i\,(h_{s(b_{\zeta}),r(c)}(|\zeta|)-h_{r(b_{\zeta}),s(c)}(|\zeta|))$
$\displaystyle\,\,\,\,=\langle
b_{\zeta},\psi(c;\zeta)\rangle_{\gamma_{k}^{\mathbb{C}}},$

where
$\psi(c;\zeta):=R_{\zeta}r(c)\circ\sqrt{2}J_{2}+i\,(R_{\zeta}s(c)\circ\sqrt{2}J_{2})$.
Consequently, we may apply the Cauchy-Schwarz inequality, and Lemma 7.1-(ii)
implies that

$|h_{b,c}(\zeta)|\leq\|b_{\zeta}\|_{\gamma_{k}^{\mathbb{C}}}\,\|\psi(c;\zeta)\|_{\gamma_{k}^{\mathbb{C}}}=\|b\|_{\gamma_{k}^{\mathbb{C}}}\,\|\psi(c;\zeta)\|_{\gamma_{k}^{\mathbb{C}}}.$

Since $\|T_{\zeta}\|_{\mathcal{L}(L^{2}(\gamma_{2k}))}=1$ (see [19, Theorem
1.4.1]), a two-fold application of (2.27) implies that

$\|\psi(c;\zeta)\|_{\gamma_{k}^{\mathbb{C}}}^{2}=\|R_{\zeta}r(c)\|_{\gamma_{2k}}^{2}+\|R_{\zeta}s(c)\|_{\gamma_{2k}}^{2}\leq\|r(c)\|_{\gamma_{2k}}^{2}+\|s(c)\|_{\gamma_{2k}}^{2}=\|c\|_{\gamma_{k}^{\mathbb{C}}}^{2}.$

In conclusion, we finally obtain

$|h_{b,c}(\zeta)|\leq\|b\|_{\gamma_{k}^{\mathbb{C}}}\,\|\psi(c;\zeta)\|_{\gamma_{k}^{\mathbb{C}}}\leq\|b\|_{\gamma_{k}^{\mathbb{C}}}\,\|c\|_{\gamma_{k}^{\mathbb{C}}}.$

∎

By taking into account that
${\text{sign}}(\zeta)\,|\zeta|^{2\nu+1}=\zeta\cdot\zeta^{\nu}\cdot\overline{\zeta}^{\;\nu}$
for all $\zeta\in\mathbb{C}$ and $\nu\in\mathbb{N}_{0}$, Theorem 7.2 also
leads to a straightforward generalisation of the Haagerup function (cf. [56,
Proof of Theorem 3.1] and Example 7.15). To this end, we introduce a class of
complex-valued functions which could be viewed as a transfer of the class of
all odd real-valued functions to the complex field and contains the complex
signum function $\text{sign}:\mathbb{C}\longrightarrow\mathbb{C}$ as element.

######  Definition 7.3.

Let $\mathbb{F}\in\\{\mathbb{R},\mathbb{C}\\}$ and $k\in\mathbb{N}$. A
function $b:\mathbb{F}^{k}\longrightarrow\mathbb{F}$ is circularly symmetric
if

$b(\alpha z)=\alpha b(z)\text{ for all }(\alpha,z)\in
S_{\mathbb{F}}\times\mathbb{F}^{k}\,.$

The set of all circularly symmetric functions is denoted by
$CS_{k}(S_{\mathbb{F}})$.

Definition 7.3 obviously implies that
$CS_{k}(S_{\mathbb{R}})=CS_{k}(\\{-1,1\\})$ coincides with the set of all odd
real functions from $\mathbb{R}^{k}$ to $\mathbb{R}$ and that
$CS_{k}(S_{\mathbb{C}})=CS_{k}(\mathbb{T})$. Moreover,
$(CS_{k}(S_{\mathbb{F}}),\circ,\text{id})$ is a monoid (i.e., a semigroup,
with unit element), where the binary operation $\circ$ is given by the
composition of functions.

######  Corollary 7.4.

Let $k\in\mathbb{N}$ and $\zeta\in\overline{\mathbb{D}}$. Let $b,c\in
L^{2}(\mathbb{C}^{k},\gamma_{k}^{\mathbb{C}})$. Suppose that $b\in
CS_{k}(\mathbb{T})$. Then

  1. (i)

$\displaystyle
h^{\mathbb{C}}_{b,c}(\zeta)={\text{sign}}(\zeta)(h_{r(b),r(c)}(|\zeta|)+h_{s(b),s(c)}(|\zeta|)+i\,(h_{s(b),r(c)}(|\zeta|)-h_{r(b),s(c)}(|\zeta|)))\,.$
(7.225)

$\displaystyle
h^{\mathbb{C}}_{b,b}(\zeta)={\text{sign}}(\zeta)H_{r(b),s(b)}(|\zeta|)=\zeta\sum_{\nu=0}^{\infty}(p_{2\nu+1}(r(b),r(b))+p_{2\nu+1}(s(b),s(b)))\zeta^{\nu}\,\overline{\zeta}^{\;\nu}\,.$
(7.226)

In particular, we have:

    1. (i-1)

$h_{b,b}^{\mathbb{C}}\big{|}_{[-1,1]}=H_{r(b),s(b)}=h_{r(b),r(b)}+h_{s(b),s(b)}$
and $h_{b,b}^{\mathbb{C}}\big{|}_{(-1,1)}\in W^{\omega}_{+}((-1,1))$.

    2. (i-2)

If $|\zeta|=1$, then
$h^{\mathbb{C}}_{b,b}(\zeta)=\zeta\,\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2}$.

    3. (i-3)

If $\zeta\in\mathbb{D}$ and ${\text{vec}}(\textbf{X},\textbf{Y})\sim
N_{2k}(0,\Sigma_{2k}(|\zeta|))$, then

$\displaystyle\begin{split}h^{\mathbb{C}}_{b,b}(\zeta)&=\frac{{\text{sign}}(\zeta)}{(2\pi)^{2k}(1-|\zeta|^{2})^{k}}\,\int_{\mathbb{R}^{2k}}\int_{\mathbb{R}^{2k}}\big{\langle}\binom{r(b)(x)}{s(b)(x)},\binom{r(b)(y)}{s(b)(y)}\big{\rangle}_{\mathbb{R}_{2}^{2}}\exp\big{(}-\frac{\|x\|^{2}+\|y\|^{2}-2|\zeta|\,\langle
x,y\rangle}{2(1-|\zeta|^{2})}\big{)}\,\textup{d}^{2k}x\,\textup{d}^{2k}y\\\
&={\text{sign}}(\zeta)(\mathbb{E}[r(b)(\textbf{X})r(b)(\textbf{Y})]+\mathbb{E}[s(b)(\textbf{X})s(b)(\textbf{Y})]).\end{split}$
(7.227)

  2. (ii)

$h^{\mathbb{C}}_{b,b}$ is bounded, and

$\displaystyle|h^{\mathbb{C}}_{b,b}(\zeta)|\leq\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2}\text{
for all }\zeta\in\overline{\mathbb{D}}.$ (7.228)

  3. (iii)

If $b\not=0$, then
$\frac{1}{\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}}}\,H_{r(b),s(b)}$ as well as
$\frac{1}{\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}}}\,h^{\mathbb{C}}_{b,b}$ are CCP
functions. Both,
$H_{r(b),s(b)}:[-1,1]\longrightarrow[-\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2},\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2}]$
and
$h^{\mathbb{C}}_{b,b}:\overline{\mathbb{D}}\longrightarrow\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2}\overline{\mathbb{D}}$
are circularly symmetric homeomorphisms. $H_{r(b),s(b)}$ is strictly
increasing and satisfies
$H_{r(b),s(b)}((-1,0))=(-\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2},0)$ and
$H_{r(b),s(b)}((0,1))=(0,\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2})$. Moreover,
$h^{\mathbb{C}}_{b,b}(\mathbb{D})=\mathbb{D}$ and

$\displaystyle(h^{\mathbb{C}}_{b,b})^{-1}(w)={\text{sign}}(w)H_{r(b),s(b)}^{-1}(|w|)\text{
for all }w\in\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2}\overline{\mathbb{D}}\,.$
(7.229)

  4. (iv)

$h_{r(b),r(b)}^{\prime}(0)>0$ or $h_{s(b),s(b)}^{\prime}(0)>0$ if and only if
$H_{r(b),s(b)}^{-1}{|}_{(-\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2},\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2})}=(H_{r(b),s(b)}{\big{|}}_{(-1,1)})^{-1}$
is real analytic on
$(-\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2},\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2})$.

###### Proof.

(i): Firstly, since $b\in CS_{k}(\mathbb{T})$, $b$ in particular is an odd
function; i.e., $b(-z)=-b(z)$ for all $z\in\mathbb{C}^{k}$. Thus, also $r(b)$
and $s(b)$ are odd (since $2\operatorname{Re}(b)=b+\overline{b}$ and
$2i\operatorname{Im}(b)=b-\overline{b}$), such as $h_{r(b),r(b)}$,
$h_{s(b),s(b)}$ and $H_{r(b),s(b)}$.
Fix $\zeta\in\overline{\mathbb{D}}$. Since $b\in CS_{k}(\mathbb{T})$, it
follows that

$h^{\mathbb{C}}_{b,c}(\zeta)={\text{sign}}(\zeta)h^{\mathbb{C}}_{b_{\overline{\zeta}},c}(\zeta)$

(which also holds for $\zeta=0$ due to (7.222)). (7.225) now follows from
Lemma 7.1-(ii), (2.27) and (7.219), where the latter is applied to the well-
defined function $\overline{\mathbb{D}}\setminus\\{0\\}\ni z\mapsto
h^{\mathbb{C}}_{b_{\overline{\zeta}},c}(z)$. In particular,

$h^{\mathbb{C}}_{b,b}(\zeta)={\text{sign}}(\zeta)H_{r(b),s(b)}(|\zeta|)$

(due to (7.220)). Therefore, Theorem 6.5, (iii), applied to the odd functions
$h_{r(b),r(b)}$ and $h_{s(b),s(b)}$ immediately lead to the power series
representation (7.226) of $h^{\mathbb{C}}_{b,b}$, which holds on
$\overline{\mathbb{D}}$. (7.227) now follows from (2.33) and (7.214).
(ii) (7.228) is an implication of (7.221).
(iii) Assume that $b\not=0$. Then
$H_{r(b),s(b)}=\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}}\,\psi_{r(b),s(b)}$, where
$\psi_{r(b),s(b)}:=\frac{H_{r(b),s(b)}}{\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}}}$
is CCP (due to (7.216) and (7.217)). (7.226), together with Theorem 5.20
(where the latter is applied to the coefficients
$a_{kl}:=(p_{2\nu+1}(r(b),r(b))+p_{2\nu+1}(s(b),s(b)))\delta_{k,l+1}$,
$k,l\in\mathbb{N}_{0}$) implies that
$\frac{1}{\|b\|^{2}_{\gamma_{k}^{\mathbb{C}}}}\,h^{\mathbb{C}}_{b,b}:\overline{\mathbb{D}}\longrightarrow\overline{\mathbb{D}}$
is a CCP function. (7.216) and Theorem 6.8-(iii) imply that both,
$H_{r(b),s(b)}$ and $h^{\mathbb{C}}_{b,b}$ are homeomorphisms. The
representation (7.229) of $(h^{\mathbb{C}}_{b,b})^{-1}$ follows at once. (i-2)
and Theorem 6.8-(iii) obviously conclude the proof of (iii).
(iv) Since $h_{r(b),r(b)}^{\prime}(0)>0$ or $h_{s(b),s(b)}^{\prime}(0)>0$ if
and only if $H_{r(b),s(b)}^{\prime}(0)>0$, and since $H_{r(b),s(b)}$ is odd,
we may adopt the underlying idea of the proof of Theorem 6.8,(iv), implying
statement (iv). ∎

######  Remark 7.5.

Already the trivial example $b:=1$, $c:=1$ and $\zeta:=i$ shows us that the
additional assumption $b\in CS_{k}(\mathbb{T})$ in Corollary 7.4 cannot be
dropped.

Recall (2.29), including the construction of the function $g^{\mathbb{C}}$
therein. Due to Proposition 6.12 and Corollary 7.4 we obtain

######  Remark 7.6.

Let $k\in\mathbb{N}$ and $f\in L^{2}(\mathbb{R}^{k},\gamma_{k})$. Then
$r(f^{\mathbb{C}})=f\otimes 1$ and $s(f^{\mathbb{C}})=0$. Moreover,

$h_{f,f}=h_{f\otimes 1,f\otimes 1}=H_{r(f^{\mathbb{C}}),s(f^{\mathbb{C}})}\,.$

If in addition $f^{\mathbb{C}}\in CS_{k}(\mathbb{T})$, then
$h_{f,f}=h^{\mathbb{C}}_{f^{\mathbb{C}},f^{\mathbb{C}}}{\big{|}}_{[-1,1]}$ .

At this point, it is very useful to recall Theorem 6.35 and its proof, where
we also implemented Abel’s theorem on real power series. From complex analysis
it is well-known that in general Abel’s theorem on power series in this form
does not hold for $\mathbb{F}=\mathbb{C}$. Moreover, already the structure of
the somewhat “simpler” complex-valued odd functions $h_{b,c}$ (cf. (7.225) and
(7.226)) seemingly does not allow a transfer of Theorem 6.35-(ii) and (iii) to
the complex case (due to Theorem 5.20). However, since [55, Lemma 1.1.13,
statement 2] also holds also for the complex field, the proof of Theorem 6.35
can be easily adapted, as well as the proof of Corollary 6.36, so that at
least the following versions of a complex hybrid correlation transform can be
stated at once:

######  Proposition 7.7.

Let $m,n\in\mathbb{N}$, $0<r<\infty$ and

$M:=\begin{pmatrix}A&S\\\ S^{\ast}&B\end{pmatrix}\in
M_{m+n}(r\,\overline{\mathbb{D}})^{+}$

be positive semidefinite, where any entry of the matrices $A,S$ and $B$ is an
element of $r\,\overline{\mathbb{D}}$. Let
$f,g:(-r,r)\longrightarrow\mathbb{R}$ be two functions, such that $(-r,r)\ni
x\mapsto f(x)=\sum_{\nu=0}^{\infty}a_{\nu}x^{\nu}\in W^{\omega}_{+}((-r,r))$
and $(-r,r)\ni x\mapsto g(x)=\sum_{\nu=0}^{\infty}b_{\nu}x^{\nu}\in
W^{\omega}_{+}((-r,r))$. Let $q:r\mathbb{D}\longrightarrow\mathbb{C}$ be a
holomorphic function, such that

$\displaystyle|c_{\nu}|\leq\sqrt{|a_{\nu}|\,|b_{\nu}|}\text{ for all
}\nu\in\mathbb{N}_{0}\,,$ (7.230)

where $c_{\nu}:=\frac{q^{(\nu)}(0)}{\nu!}$. Then $q_{r}\in W^{+}(\mathbb{D})$,
where $q_{r}(\zeta):=q(r\zeta)$ for all $\zeta\in\mathbb{D}$. Put
$r\,\overline{\mathbb{D}}\ni
z\mapsto\tilde{q}(z):=\sum_{\nu=0}^{\infty}c_{\nu}z^{\nu}$,
$r\,\overline{\mathbb{D}}\ni z\mapsto
f_{\text{abs}}(z):=\sum_{\nu=0}^{\infty}|a_{\nu}|z^{\nu}$ and
$r\,\overline{\mathbb{D}}\ni z\mapsto
g_{\text{abs}}(z):=\sum_{\nu=0}^{\infty}|b_{\nu}|z^{\nu}$. Then
$\widetilde{q_{r}}(\frac{z}{r})=\tilde{q}(z)$ for all $z\in
r\,\overline{\mathbb{D}}$ and $\tilde{q}\big{|}_{r\,\mathbb{D}}=q$, where
$\widetilde{q_{r}}\in A(\mathbb{D})$ is the continuous extension of $q_{r}$.
Moreover, the following properties hold:

  1. (i)

$\begin{pmatrix}f_{\text{abs}}[A]&\tilde{q}[S]\\\
\tilde{q}[S]^{\ast}&g_{\text{abs}}[B]\end{pmatrix}\in
M_{m+n}(r\,\overline{\mathbb{D}})^{+}$

is positive semidefinite.

  2. (ii)

If $0<c_{\ast}\leq r$ is a root of $f_{\text{abs}}-1$, then

$\tilde{q}[c_{\ast}\,\Gamma]\in\mathcal{Q}_{m,n}(\mathbb{C})\text{ for all
}\Gamma\in\mathcal{Q}_{m,n}(\mathbb{C})\,.$

Thus, an application of Lemma 5.10 immediately leads us to

######  Corollary 7.8.

Let $m,n\in\mathbb{N},A\in\mathbb{M}_{m,n}(\mathbb{C})$ and

$\Sigma=\begin{pmatrix}M&S\\\ S^{\top}&N\end{pmatrix}\in C(m+n;\mathbb{C})$

be an arbitrary complex $(m+n)\times(m+n)$ correlation matrix (with block
elements $M\in C(m;\mathbb{C}),N\in C(n;\mathbb{C})$ and
$S\in{\mathcal{Q}}_{m,n}(\mathbb{C})$). Let $r>0$ and $0\not=\psi\in
W^{\omega}_{+}((-r,r))$. Put $r\,\overline{\mathbb{D}}\ni
z\mapsto\tilde{\psi}(z):=\sum_{\nu=0}^{\infty}a_{\nu}z^{\nu}$, where
$a_{\nu}:=\frac{\psi^{(\nu)}(0)}{\nu!}$. If $0<c\leq r$, then

$\frac{1}{\psi_{\text{abs}}(c)}\begin{pmatrix}\psi_{\text{abs}}[c\,M]&\tilde{\psi}[c\,S]\\\\[5.0pt]
\tilde{\psi}[c\,S^{\ast}]&\psi_{\text{abs}}[c\,N]\end{pmatrix}\in
C(m+n;\mathbb{C})$

again is a correlation matrix with complex entries. In particular,

$\displaystyle\frac{1}{\psi_{\text{abs}}(c)}\widetilde{\psi}[cS]\in{\mathcal{Q}}_{m,n}(\mathbb{C})\,\text{
for all }\,S\in{\mathcal{Q}}_{m,n}(\mathbb{C})\,.$ (7.231)

###  7.2 On complex bivariate Hermite polynomials

Our next aim is to reveal in detail that it is possible to transfer the
content of Theorem 6.40 from the real case to the complex one, while
maintaining our constructive proof (including the intended avoidance of the
tensor product language). However, we cannot simply copy the proof of Theorem
6.40. Nevertheless, we are going to unfurl that in fact it is possible to
transfer (6.194) and (6.195) from the real case to the complex one. To this
end, we are going to work with a particular case of the rich class of complex
bivariate Hermite polynomials, first considered by K. Itô while working with
complex multiple Wiener integrals (cf. [74]). Similarly to the real case, we
need to verify a convenient correlation property of a random version of these
polynomials, which to the best of our knowledge have not been published before
(see Theorem 7.11 below). A detailed introductory discussion of complex
Hermite polynomials and their structure (which would exceed the topic of our
paper by far) can be studied in [47, 73]. Firstly, we have to recall the
following general construction:

######  Definition 7.9 (Complex Hermite polynomial).

Let $m,n\in\mathbb{N}_{0}$ and $z,w\in\mathbb{C}$. The complex Hermite
polynomial $H_{m,n}:\mathbb{C}^{2}\longrightarrow\mathbb{C}$ is defined as

$H_{m,n}(z,w):=\frac{1}{\sqrt{m!n!}}\sum_{j=0}^{m\wedge
n}(-1)^{j}j!\binom{m}{j}\binom{n}{j}z^{m-j}\,w^{n-j}\,.$

Within the scope of our work, we need the particular case of Itô’s complex
Hermite polynomials

$\displaystyle\mathbb{C}\ni z$ $\displaystyle\mapsto
H_{m,n}\circ\kappa(z):=H_{m,n}(z,\overline{z})$
$\displaystyle=i^{m+n}\sum_{j=0}^{m}\sum_{k=0}^{n}i^{j+k}(-1)^{j+n}\sqrt{\binom{m}{j}\binom{n}{k}}s(j,k)H_{j+k}(\sqrt{2}\operatorname{Re}(z))\,s(m-j,n-k)H_{m-j+n-k}(\sqrt{2}\operatorname{Im}(z)),$

where $\kappa(z):={\text{vec}}(z,\overline{z})$ and
$s(\nu,\mu):=\sqrt{\tfrac{(\nu+\mu)!}{\nu\,!\mu!}}$ for all
$\nu,\mu\in\mathbb{N}_{0}$, including the following facts, which we list
without proof (cf. [74, 73]).

######  Theorem 7.10.

Let $m,n\in\mathbb{N}_{0}$ and $z,w\in\mathbb{C}$. Then

  1. (i)

$\\{H_{m,n}\circ\kappa:m,n\in\mathbb{N}_{0}\\}$ is an orthonormal basis in the
complex Hilbert space $L^{2}(\gamma_{1}^{\mathbb{C}})$.

  2. (ii)

The exponential generating function of
$\\{H_{m,n}\circ\kappa:m,n\in\mathbb{N}_{0}\\}$ is given as

$\sum_{m,n=0}^{\infty}H_{m,n}(z,\overline{z})\frac{u^{m}}{\sqrt{m}!}\frac{v^{n}}{\sqrt{n}!}=\exp(uz+v\overline{z}-uv)\text{
for all }u,v\in\mathbb{C}.$

Lemma 2.7-(ii) allows us to transfer (the special case $k=1$ of) Corollary
6.13 to the complex case. More precisely, we have

######  Theorem 7.11.

Let $m,n,\nu,\mu\in\mathbb{N}_{0}$ and $\zeta\in\overline{\mathbb{D}}$. If
${\text{vec}}(Z,W)\sim\mathbb{C}N_{2}(0,\Sigma_{2}(\zeta))$, then

$\mathbb{E}\big{[}H_{m,n}(Z,\overline{Z})\overline{H_{\nu,\mu}(W,\overline{W})}\,\big{]}=\delta_{m,\nu}\,\delta_{n,\mu}\,\zeta^{m}\,\overline{\zeta}^{\,n}=\delta_{(m,n),(\nu,\mu)}\,\zeta^{m}\overline{\zeta}^{\,n}.$

###### Proof.

Let $u,v,a,b\in\mathbb{C}$. Due to Theorem 7.10-(ii), a multiplication of the
two (random) exponential generating functions leads to

$\sum_{m,n=0}^{\infty}\sum_{\nu,\mu=0}^{\infty}H_{m,n}(Z,\overline{Z})\overline{H_{\nu,\mu}(W,\overline{W})}\frac{u^{m}}{\sqrt{m!}}\frac{v^{n}}{\sqrt{n!}}\frac{\overline{a}^{\nu}}{\sqrt{\nu!}}\frac{\overline{b}^{\mu}}{\sqrt{\mu!}}=\exp\big{(}\binom{\overline{u}}{b}^{\ast}\binom{Z}{W}+\binom{\overline{v}}{a}^{\ast}\overline{\binom{Z}{W}}\big{)}\exp(-uv-\overline{a}\overline{b}).$

Thus, if we put
$\alpha_{m,n,\nu,\mu}(\zeta):=\mathbb{E}\big{[}H_{m,n}(Z,\overline{Z})\overline{H_{\nu,\mu}(W,\overline{W})}\,\big{]}$,
then Lemma 2.7-(ii) implies that

$\sum_{m,n=0}^{\infty}\sum_{\nu,\mu=0}^{\infty}\alpha_{m,n,\nu,\mu}(\zeta)\frac{u^{m}v^{n}\overline{a}^{\nu}\overline{b}^{\mu}}{\sqrt{m!n!\nu!\mu!}}=\exp\big{(}(u,\overline{b})\Sigma_{2}(\zeta)\binom{v}{\overline{a}}\big{)}\exp(-uv-\overline{a}\overline{b})=\exp(u\overline{a}\zeta+\overline{b}v\overline{\zeta}).$

Since $u,v,a,b\in\mathbb{C}$ were arbitrarily chosen, the multi-index notation
shows us, that we actually have proven that

$\sum_{n\in\mathbb{N}_{0}^{4}}\frac{\beta_{n}(\zeta)}{n!}(z_{1},z_{2},z_{3},z_{4})^{n}=\exp(z_{1}z_{3}\,\zeta+z_{2}z_{4}\,\overline{\zeta})=\exp(z_{1}z_{3}\,\zeta)\exp(z_{2}z_{4}\,\overline{\zeta})\text{
for all }z_{1},z_{2},z_{3},z_{4}\in\mathbb{C},$

where $\beta_{n}(\zeta):=\sqrt{n!}\,\alpha_{n}(\zeta)$. Consequently, it
follows that

$\sum_{n\in\mathbb{N}_{0}^{4}}\big{(}\frac{\beta_{n}(\zeta)-\sqrt{n!}\,\delta_{n_{1},n_{3}}\,\delta_{n_{2},n_{4}}\,\zeta^{n_{1}}\,\overline{\zeta}^{n_{2}}}{n!}\big{)}z^{n}=0\text{
for all }z\in\mathbb{C}^{4}\,,$

wherefrom the claim follows (by uniqueness of the multi-dimensional power
series expansion for the entire function $f=0$ around $0$ [124, Ch. 1.2.2.]).
∎

###  7.3 Upper bounds of $K_{G}^{\mathbb{C}}$ and inversion of complex CCP
functions

Equipped with the complex Hermite polynomials and Theorem 7.11, it is possible
to transfer Theorem 6.32 from the real field $\mathbb{R}$ to the complex field
$\mathbb{C}$; at least in the odd case. We “just” have to construct the
mappings $\alpha^{\psi,c}_{1}\in S_{L^{2}(\gamma_{1}^{\mathbb{C}})}$ and
$\beta^{\psi,c}_{1}\in S_{L^{2}(\gamma_{1}^{\mathbb{C}})}$ properly.

######  Theorem 7.12.

Let $k\in\mathbb{N}$ and $0<c\leq 1$. Let $0\not=\psi\in
W^{\omega}_{+}((-1,1))$ be odd. Then there exist
$\alpha\equiv\alpha_{\psi,c},\beta\equiv\beta_{\psi,c}\in
S_{L^{2}(\gamma_{1}^{\mathbb{C}})}$, which satisfy the following properties:

  1. (i)

$\mathbb{E}[\alpha(Z)]=\mathbb{E}[\beta(Z)]=0$ for all
$Z\sim{\mathbb{C}}N_{1}(0,1)$.

  2. (ii)

If $c\,\zeta\in\mathbb{D}$, then

${\text{sign}}(\zeta)\psi(c|\zeta|)=\psi_{\text{abs}}(c)\,h_{\alpha,\beta}^{\mathbb{C}}(\zeta)$

and

${\text{sign}}(\zeta)\psi_{\text{abs}}(c|\zeta|)=h_{\alpha,\alpha}^{\mathbb{C}}(\zeta)\psi_{\text{abs}}(c)=h_{\beta,\beta}^{\mathbb{C}}(\zeta)\psi_{\text{abs}}(c).$

In particular,

$\displaystyle\psi(c)=\psi_{\text{abs}}(c)\,\langle\alpha,\beta\rangle_{\gamma_{1}^{\mathbb{C}}}\,.$
(7.232)

  3. (iii)

If $c\not=1$ and $H$ is an arbitrary $\mathbb{C}$-Hilbert space, then

${\text{sign}}(\langle u,v\rangle_{H})\psi(c|\langle
u,v\rangle_{H}|)=\psi_{\text{abs}}(c)\,h_{\alpha,\beta}^{\mathbb{C}}(\langle
u,v\rangle_{H})$

and

${\text{sign}}(\langle u,v\rangle_{H})\psi_{\text{abs}}(c\,|\langle
u,v\rangle_{H}|)=\psi_{\text{abs}}(c)\,h_{\alpha,\alpha}^{\mathbb{C}}(\langle
u,v\rangle_{H})=\psi_{\text{abs}}(c)\,h_{\beta,\beta}^{\mathbb{C}}(\langle
u,v\rangle_{H})$

for all $u,v\in S_{H}$.

###### Proof.

Let $0<c\leq 1$ and $\zeta\in\overline{\mathbb{D}}$. Since $0\not=\psi\in
W^{\omega}_{+}((-1,1))$, it follows that $\psi^{(n_{0})}(0)\not=0$ for some
$n_{0}\in\mathbb{N}_{0}$, implying that $\psi_{\text{abs}}(x)>0$ for all
$x\in(0,1]$. Thus, $\psi_{\text{abs}}(c)>0$. Put
$\widetilde{b_{n}}\equiv\widetilde{b_{n_{1},n_{2}}}:=\frac{\psi^{(2n_{1}+1)}(0)}{(2n_{1}+1)!}\delta_{n_{1},n_{2}}$,
where $n=(n_{1},n_{2})\in\mathbb{N}_{0}^{2}$. Because of (3.92) and (3.93), we
may choose complex random variables $Z_{\zeta}\sim{\mathbb{C}}N_{1}(0,1)$ and
$Z_{1}\sim{\mathbb{C}}N_{1}(0,1)$, such that
${\text{vec}}(Z_{\zeta},Z_{1})\sim{\mathbb{C}}N_{2}(0,\Sigma_{2}(\zeta))$,
${\text{vec}}(Z_{\zeta},Z_{\zeta})\sim{\mathbb{C}}N_{2}(0,\Sigma_{2}(1))$ and
${\text{vec}}(Z_{1},Z_{1})\sim{\mathbb{C}}N_{2}(0,\Sigma_{2}(\zeta))$. Theorem
7.11 therefore in particular implies that

$\displaystyle\mathbb{E}\big{[}H_{k+1,l}(Z_{\zeta},\overline{Z_{\zeta}})\overline{H_{\nu+1,\mu}(Z_{1},\overline{Z_{1}})}\,\big{]}=\delta_{(k,l),(\nu,\mu)}\,\zeta^{k}\overline{\zeta}^{\,l}.$
(7.233)

and

$\displaystyle\mathbb{E}\big{[}H_{k+1,l}(Z_{\zeta},\overline{Z_{\zeta}})\overline{H_{\nu+1,\mu}(Z_{\zeta},\overline{Z_{\zeta}})}\,\big{]}=\delta_{(k,l),(\nu,\mu)}=\mathbb{E}\big{[}H_{k+1,l}(Z_{1},\overline{Z_{1}})\overline{H_{\nu+1,\mu}(Z_{1},\overline{Z_{1}})}\,\big{]}$
(7.234)

for all $k,l,\nu,\mu\in\mathbb{N}_{0}$. Consequently, since $\psi$ is odd by
assumption, a straightforward calculation (including the definition of the
real numbers $\widetilde{b_{n}}$ and (2.27) and (7.234)) implies that

$\mathbb{C}\ni
z\mapsto\alpha(z)\equiv\alpha_{\psi,c}(z):=\frac{1}{\sqrt{\psi_{\text{abs}}(c)}}\sum_{n\in\mathbb{N}_{0}^{2}}\text{sign}(\widetilde{b_{n}})\,\sqrt{|\widetilde{b_{n}}|}H_{n_{1}+1,n_{2}}(z,\overline{z})(\sqrt{c})^{|n|+1}$

and

$\mathbb{C}\ni
z\mapsto\beta(z)\equiv\beta_{\psi,c}(z):=\frac{1}{\sqrt{\psi_{\text{abs}}(c)}}\sum_{n\in\mathbb{N}_{0}^{2}}\sqrt{|\widetilde{b_{n}}|}H_{m_{1}+1,m_{2}}(z,\overline{z})(\sqrt{c})^{|m|+1}$

both lead to well-defined elements $\alpha\in L^{2}(\gamma_{1}^{\mathbb{C}})$
and $\beta\in L^{2}(\gamma_{1}^{\mathbb{C}})$.
(i) Let $Z\sim{\mathbb{C}}N_{1}(0,1)$. Then
$(Z,Z)^{\top}\sim{\mathbb{C}}N_{2}(0,\Sigma_{2}(1))$ (due to (2.23)). Since
$H_{0,0}=1$, Theorem 7.11 therefore implies that

$\mathbb{E}\big{[}H_{l_{1}+1,l_{2}}(Z,\overline{Z})\big{]}=\mathbb{E}\big{[}H_{l_{1}+1,l_{2}}(Z,\overline{Z})\overline{H_{0,0}(Z,\overline{Z})}\,\big{]}=0\text{
for all }(l_{1},l_{2})\in\mathbb{N}_{0}^{2},$

and (i) follows.
(ii) Similarly, as explained above, (7.233), together with the construction of
the real numbers $\widetilde{b_{n}}$ and the fact that
$\zeta^{l+1}\overline{\zeta}^{l}=|\zeta|^{2l}\zeta=\text{sign}(\zeta)|\zeta|^{2l+1}$
for all $l\in\mathbb{N}_{0}$ implies that

$\psi_{\text{abs}}(c)\mathbb{E}[\alpha(Z_{\zeta})\overline{\alpha(Z_{1})}]={\text{sign}}(\zeta)\psi_{\text{abs}}(c|\zeta|)=\psi_{\text{abs}}(c)\mathbb{E}[\beta(Z_{\zeta})\overline{\beta(Z_{1})}].$

and

$\psi_{\text{abs}}(c)\mathbb{E}[\alpha(Z_{\zeta})\overline{\beta(Z_{1})}]={\text{sign}}(\zeta)\psi(|c\,\zeta|).$

However,
$\mathbb{E}[\alpha(Z_{\zeta})\overline{\beta(Z_{1})}]=h_{\alpha,\beta}^{\mathbb{C}}(\zeta)$,
$\mathbb{E}[\alpha(Z_{\zeta})\overline{\alpha(Z_{1})}]=h_{\alpha,\alpha}^{\mathbb{C}}(\zeta)$
and
$\mathbb{E}[\beta(Z_{\zeta})\overline{\beta(Z_{1})}]=h_{\beta,\beta}^{\mathbb{C}}(\zeta)$,
which proves (ii).
(iii) We just have to apply (ii) to $\overline{\mathbb{D}}\ni\zeta:=\langle
u,v\rangle_{H}$. ∎

Recall Corollary 7.4 including the structure of $h^{\mathbb{C}}_{b,b}$,
strongly built on the odd homeomorphic real CCP function
$\frac{1}{\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2}}\,H_{r(b),s(b)}:[-1,1]\longrightarrow[-1,1]$.
Thus, if we link (7.226), (7.229) and Theorem 7.12 (where the latter is
applied to the function
$\psi:=\big{(}\frac{1}{\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2}}\,H_{r(b),s(b)}\big{)}^{-1}=H_{r(b),s(b)}^{-1}(\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2}\,\cdot)$),
we are able to prove

######  Theorem 7.13 (Complex inner product rounding).

Let $k\in\mathbb{N}$ and $b\in S_{L^{2}(\gamma_{k}^{\mathbb{C}})}$. If $b\in
CS_{k}(\mathbb{T})$ and $(H_{r(b),s(b)})^{-1}{\big{|}}_{(-1,1)}\in
W^{\omega}_{+}((-1,1))$, then
$\big{(}H_{r(b),s(b)}^{-1}\big{)}_{\text{abs}}(1)>1$, and there exist
$\alpha_{b}\in L^{2}(\gamma_{1}^{\mathbb{C}})$ and $\beta_{b}\in
L^{2}(\gamma_{1}^{\mathbb{C}})$, satisfying
$0<\langle\alpha_{b},\beta_{b}\rangle_{\gamma_{1}^{\mathbb{C}}}<1$, such that
for all $\mathbb{C}$-Hilbert spaces $H$ and $u,v\in S_{H}$ the following
properties are satisfied:

  1. (i)

$\displaystyle\langle
u,v\rangle_{H}=\frac{1}{c^{\ast}}h_{b,b}^{\mathbb{C}}(\zeta_{u,v}(b)),$
(7.235)

where $0<c^{\ast}\equiv c^{\ast}(b):=H_{r(b),s(b)}^{\text{hyp}}(1)<1$ and
$\zeta_{u,v}(b):=h_{\alpha_{b},\beta_{b}}^{\mathbb{C}}(\langle
u,v\rangle_{H})\in\mathbb{D}$.

  2. (ii)

$c^{\ast}=H_{r(b),s(b)}(\langle\alpha_{b},\beta_{b}\rangle_{\gamma_{1}^{\mathbb{C}}})=h_{b,b}^{\mathbb{C}}(\langle\alpha_{b},\beta_{b}\rangle_{\gamma_{1}^{\mathbb{C}}}).$

  3. (iii)

If ${\text{vec}}(\textbf{X},\textbf{Y})\sim
N_{2k}(0,\Sigma_{2k}(|\zeta_{u,v}(b)|))$, then

$\displaystyle c^{\ast}\,\langle u,v\rangle_{H}$
$\displaystyle=\frac{{\text{sign}}(\zeta_{u,v})}{(2\pi)^{2k}(1-|\zeta_{u,v}|^{2})^{k}}\,\int_{\mathbb{R}^{2k}}\int_{\mathbb{R}^{2k}}\big{\langle}\binom{r(b)(x)}{s(b)(x)},\binom{r(b)(y)}{s(b)(y)}\big{\rangle}_{\mathbb{R}_{2}^{2}}\exp\big{(}-\frac{\|x\|^{2}+\|y\|^{2}-2|\zeta_{u,v}(b)|\,\langle
x,y\rangle}{2(1-|\zeta_{u,v}(b)|^{2})}\big{)}\textup{d}^{2k}x\,\textup{d}^{2k}y$
$\displaystyle={\text{sign}}(\zeta_{u,v})\big{(}\mathbb{E}[r(b)(\textbf{X})r(b)(\textbf{Y})]+\mathbb{E}[s(b)(\textbf{X})s(b)(\textbf{Y})]\big{)},$

  4. (iv)

If $m,n\in\mathbb{N}$ and $(z,w)\in S_{H}^{m}\times S_{H}^{n}$, then there
exist $m+n$ $\mathbb{C}^{k}$-valued random vectors
$\textbf{Z}_{1},\ldots,\textbf{Z}_{m},\textbf{W}_{1},\ldots,\textbf{W}_{n}$,
such that
${\text{vec}}(\textbf{Z}_{i},\textbf{W}_{j})\sim{\mathbb{C}}N_{2k}(0,\Sigma_{2k}(\zeta_{ij}(b)))$
for all $(i,j)\in[m]\times[n]$, and

$\displaystyle\Gamma_{H}(z,w)=\frac{1}{c^{\ast}}\,\mathbb{E}[\overline{{\textbf{P}}_{b}}{\textbf{Q}}_{b}^{\top}]\,,$
(7.236)

where $({\textbf{P}}_{b})_{i}:=b(\textbf{Z}_{i})$,
$({\textbf{Q}}_{b})_{j}:=b(\textbf{W}_{j})$ and
$\zeta_{ij}(b):=h_{\alpha_{b},\beta_{b}}^{\mathbb{C}}(\langle
z_{i},w_{j}\rangle_{H})\in\mathbb{D}$, $(i,j)\in[m]\times[n]$.

###### Proof.

(i) Fix $u,v\in S_{H}$. Put $\psi:=H_{r(b),s(b)}^{-1}$ and $\zeta:=\langle
u,v\rangle_{H}$. Since $b\in CS_{k}(\mathbb{T})$, it follows that both, $r(b)$
and $s(b)$ are odd. Hence, the CCP function
$H_{r(b),s(b)}=h_{r(b),r(b)}+h_{s(b),s(b)}$ is odd as well (Corollary 7.4),
implying that also its inverse $\psi$ is an odd function. So, we may apply
Proposition 6.9, together with Theorem 6.33, and it follows that
$0<c^{\ast}=H_{r(b),s(b)}^{\text{hyp}}(1)<1$ is well-defined and satisfies
$\psi_{\text{abs}}(c^{\ast})=1$. (7.229), together with Theorem 7.12 therefore
implies that

$(h_{b,b}^{\mathbb{C}})^{-1}(c^{\ast}\,\zeta)={\text{sign}}(c^{\ast}\,\zeta)\psi(|c^{\ast}\,\zeta|)={\text{sign}}(\zeta)\,\psi(c^{\ast}\,|\zeta|)=\psi_{\text{abs}}(c^{\ast})\,\zeta_{u,v}(b)=\zeta_{u,v}(b),$

where $\alpha_{b}:=\alpha_{\psi,c^{\ast}},\beta_{b}:=\beta_{\psi,c^{\ast}}$
and
$\zeta_{u,v}(b):=h_{\alpha_{b},\beta_{b}}^{\mathbb{C}}(\zeta)\in\overline{\mathbb{D}}$
(due to (7.221)). Hence,
$c^{\ast}\,\zeta=h_{b,b}^{\mathbb{C}}(\zeta_{u,v}(b))$. However, since
$c^{\ast}\,\zeta\in\mathbb{D}$, it follows that $\zeta_{u,v}(b)\in\mathbb{D}$
(due to Corollary 7.4, (i)).
(ii) Since
$H_{r(b),s(b)}(0)=0<c^{\ast}<1=\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2}=H_{r(b),s(b)}(1)$
and $\psi_{\text{abs}}(c^{\ast})=1$, the strict monotonicity of the odd
function $\psi=H_{r(b),s(b)}^{-1}$ implies that
$0=\psi(0)<\psi(c^{\ast})\stackrel{{\scriptstyle\eqref{eq:psi_c_vs_psi_abs_c_complex_case}}}{{=}}\langle\alpha_{b},\beta_{b}\rangle_{\gamma_{1}^{\mathbb{C}}}<\psi(1)=1$.
Thus,
$\text{sign}(\langle\alpha_{b},\beta_{b}\rangle_{\gamma_{1}^{\mathbb{C}}})=1$,
and we obtain

$c^{\ast}=H_{r(b),s(b)}(\langle\alpha_{b},\beta_{b}\rangle_{\gamma_{1}^{\mathbb{C}}})\stackrel{{\scriptstyle\eqref{eq:Haagerup_style}}}{{=}}h_{b,b}^{\mathbb{C}}(\langle\alpha_{b},\beta_{b}\rangle_{\gamma_{1}^{\mathbb{C}}}).$

(iii) This is a direct implication of (i) and Corollary 7.4 (since
$\zeta_{u,v}(b)\in\mathbb{D}$).
(iv) Let $(z,w)\in S_{H}^{m}\times S_{H}^{n}$ and $(i,j)\in[m]\times[n]$.
Because of (i), there exists
${\text{vec}}(\textbf{Z}_{i},\textbf{W}_{j})\sim{\mathbb{C}}N_{2k}(0,\Sigma_{2k}(\zeta_{ij}(b)))$
(where $\zeta_{ij}(b)\equiv\zeta_{z_{i},w_{j}}(b)$), such that

$\displaystyle\Gamma_{H}(z,w)_{ij}$ $\displaystyle=\langle
w_{j},z_{i}\rangle_{H}=\overline{\langle
z_{i},w_{j}\rangle_{H}}\stackrel{{\scriptstyle(i)}}{{=}}\frac{1}{c^{\ast}}\,\overline{h_{b,b}(\zeta_{ij}(b))}$
$\displaystyle=\frac{1}{c^{\ast}}\,\mathbb{E}[\overline{b(\textbf{Z}_{i})}b(\textbf{W}_{j})]=\frac{1}{c^{\ast}}\,\mathbb{E}[\overline{(\textbf{P}_{b})_{i}}(\textbf{Q}_{b})_{j}]=\frac{1}{c^{\ast}}\,\mathbb{E}[\overline{\textbf{P}_{b}}\textbf{Q}_{b}^{\top}]_{ij}.$

∎

Our next result shows that in fact also Theorem 6.45, respectively Corollary
6.46 can be transferred to the complex case:

######  Theorem 7.14.

Let $k,m,n\in\mathbb{N}$ and $b,c\in L^{\infty}(\mathbb{C}^{k})$. Then $0\leq
r\equiv r_{k}(b):=\|b\|_{\gamma_{k}^{\mathbb{C}}}^{2}<\infty$ and
$\|b\|_{\infty}\geq\sqrt{r}$. Moreover, the following statements hold:

  1. (i)

$|{\text{tr}}(A^{\ast}h^{\mathbb{C}}_{b,c}[S])|\leq\|b\|_{\infty}\,\|c\|_{\infty}\,\|A\|^{\mathbb{C}}_{\infty,1}\text{
for all }A\in\mathbb{M}_{m,n}(\mathbb{C})\text{ and
}S\in{\mathcal{Q}}_{m,n}(\mathbb{C}).$

  2. (ii)

Assume that $b\in CS_{k}(\mathbb{T})\setminus\\{0\\}$ and
$H_{r(b),s(b)}^{-1}{\big{|}}_{(-r,r)}\in W^{\omega}_{+}((-r,r))$. Put
$c^{\ast}\equiv
c^{\ast}(b):=H_{\frac{r(b)}{\sqrt{r}},\frac{s(b)}{\sqrt{r}}}^{\text{hyp}}(1)$.
Then $c^{\ast}\in(0,1)$, and the following properties are satisfied:

    1. (ii-1)

$\displaystyle
r\,K_{G}^{\mathbb{C}}\leq\frac{\|b\|_{\infty}^{2}}{c^{\ast}}\,.$ (7.237)

    2. (ii-2)

Let $1\leq\kappa_{\ast}<K_{G}^{\mathbb{C}}$. If $\|b\|_{\infty}=\sqrt{r}$,
then
$0<H_{\frac{r(b)}{\sqrt{r}},\frac{s(b)}{\sqrt{r}}}^{\text{hyp}}(\kappa_{\ast})<1$
and there is exactly one number number
$\gamma^{\ast}\equiv\gamma^{\ast}(b)\in(H_{\frac{r(b)}{\sqrt{r}},\frac{s(b)}{\sqrt{r}}}^{\text{hyp}}(\kappa_{\ast}),1]$,
such that

$\displaystyle
K_{G}^{\mathbb{C}}=\big{(}H_{r(b),s(b)}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(r\,\gamma^{\ast})\leq\min\big{\\{}\frac{1}{c^{\ast}},\,\big{(}H_{r(b),s(b)}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(r)\big{\\}}\,.$
(7.238)

###### Proof.

Without loss of generality, as in the proof of the real case (Theorem 6.40),
we may assume throughout the proof that $\|b\|_{\infty}=1$ and
$\|c\|_{\infty}=1$ if $\|b\|_{\infty}>0$ and $\|c\|_{\infty}>0$ (else, we just
have to rescale the pair $(b,c)$ to the pair
$\big{(}\tfrac{1}{\|b\|_{\infty}}\,b,\tfrac{1}{\|c\|_{\infty}}\,c\big{)}$). In
particular, $\|b\|_{\infty}^{2}=1$.
(i) Because of (7.213), we may fully adopt the proof of Theorem 6.40-(i).
(ii) Recall from Corollary 7.4 that both,
$H_{r(b),s(b)}:[-1,1]\longrightarrow[-r,r]$ and
$h^{\mathbb{C}}_{b,b}:\overline{\mathbb{D}}\longrightarrow
r\overline{\mathbb{D}}$ are circularly symmetric homeomorphisms. Put
$\psi_{b}:=H_{r(b),s(b)}^{-1}:[-r,r]\longrightarrow[-1,1]$ and
$a_{m}:=\frac{|\psi_{b}^{(m)}(0)|}{m!}$, $m\in\mathbb{N}$.
(ii-1) Given our assumptions, we may apply Theorem 7.13 to the function
$H_{r(b^{\circ}),s(b^{\circ})}^{-1}=\big{(}\frac{1}{r}\,H_{r(b),s(b)}\big{)}^{-1}=\psi_{b}(r\,\cdot)$,
where $b^{\circ}:=\frac{b}{\sqrt{r}}\in S_{L^{2}(\gamma_{k}^{\mathbb{C}})}$ is
an element of the unit sphere of $L^{2}(\gamma_{k}^{\mathbb{C}})$. Put
$\alpha^{\ast}:=rc^{\ast}$. Thus, if $S\in{\mathcal{Q}}_{m,n}(\mathbb{C})$ is
arbitrarily given, then

$S\stackrel{{\scriptstyle\eqref{eq:rep_of_c_times_QC_complex_case}}}{{=}}\frac{1}{c^{\ast}}\,\mathbb{E}\big{[}\overline{{\textbf{P}}_{b^{\circ}}}{\textbf{Q}}_{b^{\circ}}^{\top}\big{]}=\frac{1}{\alpha^{\ast}}\,\mathbb{E}\big{[}\overline{{\textbf{P}}_{b}}{\textbf{Q}}_{b}^{\top}\big{]},$

where the $m$-dimensional complex random vector ${\textbf{P}}_{b}$ maps into
$\overline{{\mathbb{D}}}^{\,m}$ and the $n$-dimensional complex random vector
${\textbf{Q}}_{b}$ maps into ${\overline{\mathbb{D}}}^{\,n}$ (since
$\|b\|_{\infty}=1$). Hence,

$\displaystyle|{\text{tr}}(A^{\ast}S)|=\frac{1}{\alpha^{\ast}}|{\text{tr}}(A^{\ast}\mathbb{E}[\overline{{\textbf{P}}_{b}}\,{\textbf{Q}}_{b}^{\top}])|\leq\frac{1}{\alpha^{\ast}}\,\mathbb{E}[|{\text{tr}}(A^{\ast}\overline{{\textbf{P}}_{b}}{\textbf{Q}}_{b}^{\top})|]\leq\frac{1}{\alpha^{\ast}}\,\|A\|_{\infty,1},$

and (7.237) follows.
(ii-2) Since $r=\|b\|_{\infty}^{2}=1$ (by assumption), it follows that
$\sum_{\nu=0}^{\infty}a_{2\nu+1}=(\psi_{b}\big{|}_{(-1,1)})_{\text{abs}}(1)<\infty$.
So, we may apply Lemma 5.10 to the odd real analytic function
$\psi_{b}{\big{|}}_{(-1,1)}$, whence

$\psi_{b}(|w|)=\sum_{\nu=0}^{\infty}a_{2\nu+1}{|w|}^{2\nu+1}=|w|\,\sum_{\nu=0}^{\infty}a_{2\nu+1}\,w^{\nu}\,{\overline{w}}^{\,\nu}\text{
for all }w\in\overline{\mathbb{D}}=h_{b,b}^{\mathbb{C}}(\mathbb{D}).$

Thus, (7.229) implies that

$\overline{\mathbb{D}}\ni(h_{b,b}^{\mathbb{C}})^{-1}(w)={\text{sign}}(w)\psi_{b}(|w|)=w\,\sum_{\nu=0}^{\infty}a_{2\nu+1}\,w^{\nu}\,{\overline{w}}^{\,\nu}\text{
for all }w\in\overline{\mathbb{D}}\,.$

Consequently, the matrix equality (7.213) leads to

$S=(h_{b,b}^{\mathbb{C}})^{-1}[h_{b,b}^{\mathbb{C}}[S]]=(h_{b,b}^{\mathbb{C}})^{-1}[\mathbb{E}[{\mathbf{\Lambda}}_{b}]]=\sum_{\nu=0}^{\infty}a_{2\nu+1}\mathbb{E}[{\mathbf{\Lambda}}_{b}]^{\ast\,\nu}\ast\mathbb{E}[\overline{{\mathbf{\Lambda}}_{b}}]^{\ast\,\nu}\ast\mathbb{E}[{\mathbf{\Lambda}}_{b}],$

where $\ast$ denotes entrywise multiplication on $M_{n}(\mathbb{F})$ (i.e.,
the Hadamard product of matrices) and
${\mathbf{\Lambda}}_{b}:=\overline{\textbf{R}_{b}}\,{\textbf{S}_{b}}^{\top}$.
Put
$M_{\nu}:=\mathbb{E}[{\mathbf{\Lambda}}_{b}]^{\ast\,\nu}\ast\mathbb{E}[\overline{{\mathbf{\Lambda}}_{b}}]^{\ast\,\nu}=\overline{M_{\nu}}$.
Since $\|b\|_{\infty}=1$, we may apply Lemma 6.39 to the random matrix
${\mathbf{\Lambda}}_{b}$, and it follows that

$|{\text{tr}}(A^{\ast}S)|\leq\sum_{\nu=0}^{\infty}a_{2\nu+1}\,|\langle
M_{\nu}\ast\mathbb{E}[{\mathbf{\Lambda}}_{b}],A\rangle_{2}|=\sum_{\nu=0}^{\infty}a_{2\nu+1}\,|\langle\mathbb{E}[{\mathbf{\Lambda}}_{b}],A\ast
M_{\nu}\rangle_{2}|\leq\sum_{\nu=1}^{\infty}a_{2\nu+1}\,\|A\ast
M_{\nu}\|_{\infty,1}\,,$

whereby the last inequality follows from the construction of the random Gram
matrix ${\mathbf{\Lambda}}_{b}$. Next, we may apply Lemma 6.39 twice, so that

$\|A\ast
M_{\nu}\|_{\infty,1}\leq\|A\ast\mathbb{E}[{\mathbf{\Lambda}}_{b}]^{\ast\,\nu}\|_{\infty,1}\leq\|A\|_{\infty,1}.$

Altogether, it follows that

$|{\text{tr}}(A^{\ast}S)|\leq\|A\|_{\infty,1}\,\sum_{\nu=0}^{\infty}a_{2\nu+1}=(\psi_{b}|_{(-1,1)})_{\text{abs}}(1)\,\|A\|_{\infty,1}\,.$

Consequently,
$K_{G}^{\mathbb{C}}\leq(\psi_{b}|_{(-1,1)})^{\text{abs}}(1)=\big{(}H_{r(b),s(b)}^{-1}{\big{|}}_{(-r,r)}\big{)}_{\text{abs}}(r)$.
Hence, if we put
$\gamma^{\ast}:=H_{\frac{r(b)}{\sqrt{r}},\frac{s(b)}{\sqrt{r}}}^{\text{hyp}}(K_{G}^{\mathbb{C}})$,
then $0<H_{\frac{r(b)}{\sqrt{r}},\frac{s(b)}{\sqrt{r}}}^{\text{hyp}}(1)\leq
H_{\frac{r(b)}{\sqrt{r}},\frac{s(b)}{\sqrt{r}}}^{\text{hyp}}(\kappa_{\ast})<\gamma^{\ast}\leq
1$, and (ii-1) concludes the proof of (7.238). ∎

Recall again from Theorem 5.20 that in general complex CCP functions need not
be analytic. Due to this fact and the structure of the complex-valued
(circularly symmetric) functions $h_{b,b}$, which is built on absolute values
and signs of complex numbers (cf. Corollary 7.4, including (7.226)), it seems
that Proposition 6.47 cannot be easily transferred to the complex case (if at
all).

######  Example 7.15 (Haagerup function).

Both, Corollary 7.4 and Proposition 6.16 enable us to recover quickly a direct
power series representation of Haagerup’s CCP function
$h_{b,b}:\mathbb{D}\longrightarrow\mathbb{D}$, where $b:={\text{sign}}$.
(Retranslated into the language of Haagerup, $h_{b,b}=\Phi$ and
$H_{r(b),s(b)}=\varphi$ (cf. [56, Lemma 3.5 and Theorem 3.1])). To this end,
let $0\not=\zeta\in\overline{\mathbb{D}}$ and $\rho:=|\zeta|$. As shown on
[56, p. 200], the function $h_{b,b}$ can then be written in terms of
${\text{sign}}(\zeta)$ and the two complete elliptic integrals $E(\rho)$ and
$K(\rho)$. However, in our approach (by which Haagerup’s CCP function is
obtained as a special case), we don’t have to work with elliptic integration.
Firstly note that

$r(b)(x_{1},x_{2})=r(b)(x)=\frac{x_{1}}{\|x\|_{2}}\text{ and
}s(b)(x)=r(b)(x_{2},x_{1})=\frac{x_{2}}{\|x\|_{2}}$

for all $x={\text{vec}}(x_{1},x_{2})\in\mathbb{R}^{2}\setminus{\\{0\\}}$.
Thus, if $\tau\in[-1,1]$ and ${\text{vec}}(\textbf{X},\textbf{Y})\sim
N_{2k}(0,\Sigma_{2k}(\tau))$, $k:=2$ are given, then

$\displaystyle H_{r(b),s(b)}(\tau)$
$\displaystyle=h_{r(b),r(b)}(\tau)+h_{s(b),s(b)}(\tau)$
$\displaystyle=\mathbb{E}[r(b)(\textbf{X})r(b)(\textbf{Y})]+\mathbb{E}[s(b)(\textbf{X})s(b)(\textbf{Y})]$
$\displaystyle=\mathbb{E}[r(b)(X_{1},X_{2})r(b)(Y_{1},Y_{2})]+\mathbb{E}[r(b)(X_{2},X_{1})r(b)(Y_{2},Y_{1})]$
$\displaystyle=\mathbb{E}\big{[}\frac{X_{1}\,Y_{1}}{\|\textbf{X}\|_{2}\,\|\textbf{Y}\|_{2}}\big{]}+\mathbb{E}\big{[}\frac{X_{2}\,Y_{2}}{\|\textbf{X}\|_{2}\,\|\textbf{Y}\|_{2}}\big{]}$
$\displaystyle=\mathbb{E}\big{[}\big{\langle}\frac{\textbf{X}}{\|\textbf{Y}\|_{2}},\frac{\textbf{Y}}{\|\textbf{X}\|_{2}}\big{\rangle}\big{]}$
$\displaystyle=\frac{\pi}{4}\,\tau\,{}_{2}F_{1}\big{(}\frac{1}{2},\frac{1}{2},2;\tau^{2}\big{)}=h_{f_{2},f_{2}}(\tau),$

whereby the equalities in the last line follow from Proposition 6.16 (applied
to $k=2$). An application of Corollary 7.4 (or the original proof of [56,
Theorem 3.1]) therefore implies

${\text{sign}}(\overline{\zeta})h_{b,b}(\zeta)=H_{r(b),s(b)}(\rho)=\frac{\pi}{4}\,\rho\,{}_{2}F_{1}\big{(}\frac{1}{2},\frac{1}{2},2;\rho^{2}\big{)}=h_{f_{2},f_{2}}(\rho).$

Hence,

$h_{b,b}(\zeta)={\text{sign}}(\zeta)\,\mathbb{E}\big{[}\big{\langle}\frac{\textbf{X}}{\|\textbf{X}\|_{2}},\frac{\textbf{Y}}{\|\textbf{X}\|_{2}}\big{\rangle}\big{]}=\frac{\pi}{4}\,\zeta\,{}_{2}F_{1}\big{(}\frac{1}{2},\frac{1}{2},2;|\zeta|^{2}\big{)}={\text{sign}}(\zeta)\,h_{f_{2},f_{2}}(|\zeta|)\,.$

A highly non-trivial part in [56] consists of a multi-page proof of the fact
that $h_{f_{2},f_{2}}^{-1}=H_{r(b),s(b)}^{-1}\in W^{\omega}_{+}((-1,1))$,