# Large deviation for small noise path-dependent stochastic differential
equations

Liu Xiangdong∗<EMAIL_ADDRESS>Hong Shaopeng<EMAIL_ADDRESS>
###### Abstract

In this paper, we study the asymptotic behavior of randomly perturbed path-
dependent stochastic differential equations with small parameter
$\vartheta_{\varepsilon}$, when $\varepsilon\rightarrow 0$,
$\vartheta_{\varepsilon}$ goes to $0$. When $\varepsilon\rightarrow 0$, we
establish large deviation principle. The proof of the results relies on the
weak convergence approach. As an application, we establish the large deviation
for functionals of path-dependent SDEs in small time intervals.

###### keywords:

Path-dependent stochastic differential equations , Large deviation principle ,
Weak convergence

###### MSC:

[2010] 60H10, 60F05, 60F10

organization=Department of Statistic and Data Science, School of
Economics,addressline=Jinan University, city=Guangzhou, postcode=510632,
state=Guangdong, country=PR China

##  1 Introduction

This paper sheds new light on the asymptotic behaviour of the class of path-
dependent stochastic differential equations (PSDEs).

$X(t)=X_{0}+\int_{0}^{t}b(s,X_{s})ds+\int_{0}^{t}\sigma(s,X_{s})dW(s)\quad
t\in[0,T]$ (1)

PSDEs have received increasing attentions by researchers which are much more
involved than classical SDEs as the drift and diffusion coefficients depending
on path of solution. In a nutshell, this kind of equations plays an important
role in characterising non-Markov partial differential equations (PDEs for
short). Ekren et al. [6] obtained the viscosity solutions of path-dependent
semi-linear parabolic PDEs using backward PSDEs and Non-anticipative analysis
[5, 3], and subsequently extended the results to fully nonlinear forms of
path-dependent PDEs [7].

It is well known that the key point of large deviation principle (LDP for
short) is to show the probability property of rare events. Small noise LDP for
SDEs has a long history. The pioneering work of [8] considered rare events
induced by Markov diffusions. Recently, an important contribution by [1] was
to use the weak convergence method to obtain a significant simplified
approach. Their approach avoided proof exponential continuity and tightness
estimates. Weakly convergent methods are widely used in proving large
deviations of stochastic differential equations and stochastic partial
differential equations, see [11, 13, 14] and references therein.

There have been some studies on large deviations of path-dependent SDEs. For
example, Gao and Liu [9] stuided such a problem via the sample path LDP method
by Freidlin-Wentzell and show the LDP under (r,q)-capacity. And Ma et al. [12]
based on PDEs method get the LDP of path-dependent SDEs. In this paper, we use
a different line of argument, adapting the weak convergence approach of
Budhiraja and Dupuis [1] to the path-dependent case.

Compared with the results mentioned above, the contribution of this paper is
to study LDP when the coefficients of PSDEs are all depending on
$\varepsilon$, i.e., the solutions of PSDEs possibly degenerate. As an
application, we establish the large deviation for functionals of PSDEs in
small time intervals.

The paper is organized as follows. In Section 2, we state the weak convergence
method for the large deviation principle given in Budhiraja and Dupuis [1]. We
give the main theorem and prove it in Section 3. Finally, in Section 4, we
show the large deviation principle for the functional of PSDEs in small time
interval.

We end this section with some notations. We consider a fixed time horizon
$T>0$, and denote $\mathbb{T}:=[0,T]$. Let $C([0,T];\mathbb{R}^{d})$ be the
Banach space of continuous functions $\psi:[0,T]\rightarrow\mathbb{R}^{d}$
equipped with the sup-norm $\|\psi\|:=\sup_{t\in[0,T]}|\psi(t)|$,
$\mathcal{C}^{1}_{0}([0,T];\mathbb{R}^{d})$ as the space of continuous
functions on $[0,T]$ with initial value 0 and has first-order derivative,
$\mathcal{C}^{1}_{b}([0,T];\mathbb{R}^{d})$as the space of continuous
functions on $[0,T]$ with initial value 0, has first-order derivative and has
a bound. $L^{2}$ stands short for $L^{2}(\mathbb{T})$ and $\|\cdot\|_{2}$ is
the usual $L^{2}$ norm.

##  2 Preliminaries

###  2.1 Framework

We consider small-noise convolution PSDEs

$X^{\varepsilon}(t)=X^{\varepsilon}_{0}+\int_{0}^{t}b_{\varepsilon}(s,X^{\varepsilon}_{s})ds+\vartheta_{\varepsilon}\int_{0}^{t}\sigma_{\varepsilon}(s,X^{\varepsilon}_{s})dW(s)\quad
t\in[0,T]$ (2)

taking values in $\mathbb{R}^{d}$ with $d\geq 1$, where $\varepsilon>0$ and
$\vartheta_{\varepsilon}>0$ tends to zero as $\varepsilon$ goes to zero. For
each $\varepsilon>0$, $X^{\varepsilon}_{0}\in\mathbb{R}^{d}$,
$b_{\varepsilon}:\mathbb{T}\times\mathcal{C}\left(\mathbb{T},\mathbb{R}^{d}\right)\rightarrow\mathbb{R}^{d}$,
$\sigma_{\varepsilon}:\mathbb{T}\times\mathcal{C}\left(\mathbb{T},\mathbb{R}^{d}\right)\rightarrow\mathbb{R}^{d\times
m}$are two product measurable maps that are non-anticipative in the sense that
they satisfy $b_{\varepsilon}(t,x)=b_{\varepsilon}(t,x_{t})$ and
$\sigma_{\varepsilon}(t,x)=\sigma_{\varepsilon}(t,x_{t})$ for all
$t\in\mathbb{T}$ and each
$x\in\mathcal{C}\left(\mathbb{T},\mathbb{R}^{d}\right)$, where $x_{t}$ denote
the path $x$ stopped at time $t$. $W(s)$ is an m-dimensional Brownian motion
on the filtered probability space
$\left(\Omega,\mathcal{F},\left\\{\mathcal{F}_{t}\right\\}_{t\in\mathbb{T}},\mathbb{P}\right)$
satisfying the usual conditions. We make following assumptions about the
coefficients:

  1. A.1

$X^{\varepsilon}_{0}$ converges to $x_{0}\in\mathbb{R}^{d}$ as $\varepsilon$
tends to zero.

  2. A.2

For all $\varepsilon>0$ small enough, the coefficients $b_{\varepsilon}$ and
$\sigma_{\varepsilon}$ are measurable maps on
$\mathbb{T}\times\mathcal{C}\left(\mathbb{T}:\mathbb{R}^{d}\right)$ and
converge pointwise to $b$ and $\sigma$ as $\varepsilon$ goes to zero.
Moreover, $b(t,\cdot)$ and $\sigma(t,\cdot)$ are continuous on
$\mathbb{R}^{d}$, uniformly in $t\in\mathbb{T}$.

  3. A.3

For all $\varepsilon>0$ small enough, $b_{\varepsilon}$ and
$\sigma_{\varepsilon}$ have linear growth uniformly in $\varepsilon$ and in
$t\in\mathbb{T}$. For some $L>0$

$|b_{\varepsilon}(t,\omega)|+|\sigma_{\varepsilon}(t,\omega)|\leq
M\left(1+\sup_{s\leq t}|\omega(s)|+|t|\right)$ (3)

  4. A.4

For all $\varepsilon>0$ small enough, the coefficients $b_{\varepsilon}$ and
$\sigma_{\varepsilon}$ are locally Lipschitz continuous. For any $R>0$, there
exists $L_{R}>0$ such that, for all

$|b_{\varepsilon}(t,\omega)-b_{\varepsilon}(t,\omega^{\prime})|+|\sigma_{\varepsilon}(t,\omega)-\sigma_{\varepsilon}(t,\omega^{\prime})|\leq
L_{R}(\sup_{s\leq t}|\omega(s)-\omega^{\prime}(s)|)$ (4)

###  2.2 Abstract sufficient conditions for large deviations

######  Defination 1 (Large deviation [4])

A family $\left\\{X^{\varepsilon}\right\\}_{\varepsilon>0}$ of
$\mathcal{E}$-valued random variable is said to satisfy the large deviation
principle on $\mathcal{E}$, with the good rate function $I$ and with the speed
function $\lambda(\varepsilon)$ which is a sequence of positive numbers
tending to $+\infty$ as $\varepsilon\rightarrow 0$, if the following
conditions hold:

  1. 1.

for each $M<\infty$, the level set $\\{x\in\mathcal{E}:I(x)\leq M\\}$ is a
compact subset of $E$;

  2. 2.

for each closed subset $F$ of $\mathcal{E},\limsup_{\varepsilon\rightarrow
0}\frac{1}{\lambda(\varepsilon)}\log\mathbb{P}\left(X^{\varepsilon}\in
F\right)\leq-\inf_{x\in F}I(x)$;

  3. 3.

for each open subset $G$ of $\mathcal{E},\liminf_{\varepsilon\rightarrow
0}\frac{1}{\lambda(\varepsilon)}\log\mathbb{P}\left(X^{\varepsilon}\in
G\right)\geq-\inf_{x\in G}I(x)$.

We recall here several results from Budhiraja and Dupuis [1] which gives an
abstract framework of LDP.

Let $\mathcal{A}$ denote the class of real-valued
$\left\\{\mathcal{F}_{t}\right\\}$-predictable processes $\nu$ belonging to
$L^{2}$ a.s. For each $N$ the spaces of bounded deterministic and stochastic
controls

$S_{N}:=\left\\{\nu\in L^{2};\int_{0}^{T}|{\nu}(s)|^{2}ds\leq N\right\\}.$

$S_{N}$ is endowed with the weak topology induced from
$L^{2}(\mathbb{T}\times\Omega)$. Define

$\mathcal{A}_{N}:=\left\\{\nu\in\mathcal{A};\nu(s)\in
S_{N},\mathbb{P}\text{-a.s. }\right\\}.$

######  Theorem 2 (Budhiraja and Dupuis [1])

For any $\varepsilon>0$, let $\mathcal{G}^{\varepsilon}$ be a measurable
mapping from $C([0,T];\mathbb{R})$ into $E$. Suppose that
$\left\\{\mathcal{G}^{\varepsilon}\right\\}_{\varepsilon>0}$ satisfies the
following assumptions: there exists a measurable map
$\mathcal{G}^{0}:C([0,T];\mathbb{R})\longrightarrow\mathcal{E}$ such that

  1. 1.

for every $N<+\infty$ and any family
$\left\\{\nu^{\varepsilon};\varepsilon>0\right\\}\subset\mathcal{A}_{N}$
satisfying that $\nu^{\varepsilon}$ converge in distribution as $S_{N}$-valued
random elements to $\nu$ as $\varepsilon\rightarrow
0,\mathcal{G}^{\varepsilon}\left(W.+\frac{1}{\sqrt{\varepsilon}}\int_{0}^{\cdot}\nu^{\varepsilon}(s)ds\right)$
converges in distribution to
$\mathcal{G}^{0}\left(\int_{0}^{\cdot}\nu(s)ds\right)$ as
$\varepsilon\rightarrow 0$;

  2. 2.

for every $N<+\infty$, the set
$\left\\{\mathcal{G}^{0}\left(\int_{0}^{\cdot}\nu(s)ds\right);h\in
S_{N}\right\\}$ is a compact subset of $E$.

Then the family
$\left\\{\mathcal{G}^{\varepsilon}(W(\cdot))\right\\}_{\varepsilon>0}$
satisfies a large deviation principle with the good rate function I given by

$I(g):=\inf_{\left\\{\nu\in\mathcal{H};g=\mathcal{G}^{0}\left(\int_{0}^{\cdot}\nu(s)ds\right)\right\\}}\left\\{\frac{1}{2}\int_{0}^{T}|\nu(s)|^{2}ds\right\\}\quad\text{
for }g\in\mathcal{E},$

with the convention $\inf\emptyset=\infty$.

##  3 Main Result and Proof

If A.5 hold, define the functional $\mathcal{G}$ as the Borel-measurable map
associating the multidimensional Brownian motion $W$ to the solution of the
path dependent stochastic differential systems (2), that is:
$\mathcal{G}^{\varepsilon}\left(W\right)=X^{\varepsilon}(t)$. For any control
$\nu\in\mathcal{A}_{N}$, $N>0$ and any $\varepsilon>0$, the process
$\widetilde{W}=W+\vartheta_{\varepsilon}^{-1}\int_{0}^{\cdot}\nu(s)ds$ is a
$\widetilde{\mathbb{P}}-$Brownian motion by Girsanov’s theorem, where

$\frac{d\widetilde{\mathbb{P}}}{d\mathbb{P}}:=\exp\left\\{-\frac{1}{\vartheta_{\varepsilon}}\sum_{i=1}^{m}\int_{0}^{T}\nu^{(i)}(s)dW^{(i)}(s)-\frac{1}{2\vartheta_{\varepsilon}^{2}}\int_{0}^{T}|v(s)|^{2}ds\right\\}.$
(5)

Hence the shifted version
$X^{\varepsilon,v}:=\mathcal{G}^{\varepsilon}(\tilde{W})$ appearing in Theorem
2 (1) is the strong unique solution of (2) under $\widetilde{\mathbb{P}}$,
with $X^{\varepsilon}$ and $W$ replaced by $X^{\varepsilon,v}$ and
$\widetilde{W}$. Because $\mathbb{P}$ and $\widetilde{\mathbb{P}}$ are
equivalent, $X^{\varepsilon,v}$ is also the unique strong solution, under
$\mathbb{P}$, of the controlled equation

$X^{\varepsilon,v}(t)=X_{0}^{\varepsilon}+\int_{0}^{t}\left[b_{\varepsilon}\left(s,X_{s}^{\varepsilon,v}\right)+\sigma_{\varepsilon}\left(s,X_{s}^{\varepsilon,v}\right)v(s)\right]\mathrm{d}s+\vartheta_{\varepsilon}\int_{0}^{t}\sigma_{\varepsilon}\left(s,X_{s}^{\varepsilon,v}\right)\mathrm{d}W(s)$
(6)

Taking $\varepsilon\rightarrow 0$, the system (6) reduces to the deterministic
path dependent ODE

$\phi(t)=x_{0}+\int_{0}^{t}\left[b(s,\phi_{s})+\sigma(s,\phi_{s})\nu(s)\right]ds.$
(7)

###### Theorem 3

Under A.1-A.4, the family $\left\\{X^{\varepsilon}\right\\}_{\varepsilon>0}$,
unique solution of (2), satisfies a Large Deviations Principle with rare
function $I$ and speed $\vartheta_{\varepsilon}^{-2}$, where $\mathcal{G}^{0}$
is the solution of (7)

###### Remark 1

Theorem 3 generalizes the results in Chiarini and Fischer [2]. When the
coefficients $b_{\varepsilon}$ and $\sigma_{\varepsilon}$ do not depend on the
path of the process $X_{\varepsilon}$, Theorem 3 and Chiarini and Fischer [2,
Theorem 3] are equivalent.

We first show the unique solution of (7) and a uniform estimation.

###### Lemma 4

Under A.1-A.6, given any $\nu\in L^{2}$, there is a unique solution
$\phi\in\mathcal{C}\left([0,T];\mathbb{R}^{n}\right)$ of (7). Moreover, for
$\phi$, we have the growth estimate

$\sup_{0\leq s\leq
t}\left|\phi(s)\right|\leq(3|x_{0}|^{2}+9M^{2}t(t+\|\nu\|^{2})+3M^{2}t^{3}(t+\|\nu\|^{2}))e^{9M^{2}(t+\|\nu\|^{2})}$
(8)

###### Proof 1

Let $\phi,\varphi\in\mathcal{C}\left([0,T];\mathbb{R}^{d}\right)$ be solutions
of (7). We have

$\displaystyle\left|\phi(t)-\varphi(t)\right|\leq\int_{0}^{t}\left|b(s,\phi_{s})-b(s,\varphi_{s})\right|ds+\int_{0}^{t}\left|\sigma(s,\phi_{s})-\sigma(s,\varphi_{s})\right|\left|\nu\right|ds$
(9)

By assumption A.4, we have for large enough $R>0$

$\left|\phi(t)-\varphi(t)\right|^{2}\leq
2L^{2}_{R}\left(T+\|\nu\|^{2}\right)\int_{0}^{t}\sup_{0\leq u\leq
s}\left|\phi(u)-\phi(s)\right|^{2}ds$

Gronwall’s inequality now entails that $\|\phi(t)-\varphi(t)\|=0$, which
yields uniqueness.

By using assumption A.3, we can get that

$\displaystyle\left|\phi(t)\right|^{2}$ $\displaystyle\leq
3|x_{0}|+3t\int_{0}^{t}\left|b(s,\phi_{s})^{2}\right|ds+3\left(\int_{0}^{t}\left|\sigma(s,\phi_{s})\right||\nu|ds\right)^{2}$
(10) $\displaystyle\leq
3|x_{0}|^{2}+9M^{2}\left(t+\|\nu\|^{2}\right)\int_{0}^{t}\left(1+\sup_{0\leq
u\leq s}|\phi(u)|^{2}+|s|^{2}\right)ds$ $\displaystyle\leq
3|x_{0}|^{2}+9M^{2}t(t+\|\nu\|^{2})+3M^{2}t^{3}(t+\|\nu\|^{2})+9M^{2}(t+\|\nu\|^{2})\int_{0}^{t}\sup_{0\leq
u\leq s}|\phi(u)|^{2}du$

By Gronwalls’ inequality, we can deduce

$\sup_{0\leq s\leq
t}\left|\phi(s)\right|\leq(3|x_{0}|^{2}+9M^{2}t(t+\|\nu\|^{2})+3M^{2}t^{3}(t+\|\nu\|^{2}))e^{9M^{2}(t+\|\nu\|^{2})}$

We need some technical preliminary results.

###### Lemma 5

Under A.1-A.6, for all $p\geq 2$, $N>0$, $\nu\in\mathcal{A}_{N}$ and
$\varepsilon>0$ small enough, there exists a constant $c>0$ independent of
$\varepsilon$, $\nu$, $t$ such that

$\mathbb{E}\left[\sup_{t\in\mathbb{T}}|X^{\varepsilon,\nu}(t)|^{p}\right]\leq
c$ (11)

###### Proof 2

Let us fix $p\geq 2$, $N>0$, $\nu\in\mathcal{A}_{N}$ and $t\in\mathbb{T}$. Let
$\tau_{n}$ be the stopping time defined by

$\tau_{n}=\inf\left\\{t\geq 0:|X^{\varepsilon,\nu}(t)|\geq n\right\\}\wedge T$

We write
$b^{n}_{s}:=b_{\varepsilon}(s,X^{\varepsilon,\nu}_{s}\mathbbm{1}_{\left\\{s\leq\tau_{n}\right\\}})$
and
$\sigma^{n}_{s}:=\sigma_{\varepsilon}\left(s,X^{\varepsilon,\nu}_{s}\mathbbm{1}_{\left\\{s\leq\tau_{n}\right\\}}\right)$.

We fix $n\in\mathbb{N}$ and observe that, almost surely:

$\displaystyle\mathbb{E}\left[\|X^{\varepsilon,\nu}(t)\mathbbm{1}_{t\leq\tau_{n}}\|^{p}\right]$
$\displaystyle\leq
4^{p-1}\left|X^{\varepsilon}_{0}\right|^{p}+4^{p-1}\mathbb{E}\left\\{\left[\int_{0}^{t}b_{s}^{n}ds\right]^{p}\right\\}+4^{p-1}\mathbb{E}\left\\{\left[\int_{0}^{t}\sigma_{s}^{n}\nu(s)ds\right]^{p}\right\\}$
(12)
$\displaystyle+4^{p-1}\vartheta_{\varepsilon}^{p}\mathbb{E}\left\\{\left[\int_{0}^{t}\sigma^{n}_{s}dW(s)\right]^{p}\right\\}$
$\displaystyle=:4^{p-1}\left(\left|X_{0}^{\varepsilon}\right|^{p}+I_{1}+I_{2}+I_{3}\right)$

For $\varepsilon$ small enough we can bound $\left|X^{\varepsilon}_{0}\right|$
by $2\left|X_{0}\right|$ and $\vartheta_{\varepsilon}$ by $1$. Using Holder’s
and Jensen’s inequalities, we obtain the following estimates almost surely:

$\displaystyle I_{1}$
$\displaystyle\leq\mathbb{E}\left\\{\left[\int_{0}^{t}(b_{n}^{s})^{p}ds\right]\right\\}$
(13)

and

$\displaystyle I_{2}$ $\displaystyle\leq
N^{\frac{p}{2}}\mathbb{E}\left\\{\left[\int_{0}^{t}(\sigma_{s}^{n})^{2}ds\right]^{\frac{p}{2}}\right\\}\leq
N^{\frac{p}{2}}\mathbb{E}\left\\{\int_{0}^{t}\left(\sigma_{s}^{n}\right)^{p}ds\right\\}$
(14)

By Burkholder-Davis-Gundy (B-D-G) inequality, there exists $C_{p}>0$ such that

$\mathbb{E}\left[I_{3}\right]\leq
C_{p}\mathbb{E}\left\\{\int_{0}^{t}(\sigma_{s}^{n})^{P}ds\right\\}$ (15)

From the linear growth condition on $b_{\varepsilon}$ and
$\sigma_{\varepsilon}$ we deduce that there exists $C_{1}>0$ independent of
$\varepsilon$, $\nu$, $n$ and $t$ such that for all $n\in\mathbb{N}$

$\mathbb{E}\left[\|X^{\varepsilon,\nu}(t)\mathbbm{1}_{t\leq\tau_{n}}\|^{p}\right]\leq
C_{1}+C_{1}\int_{0}^{t}\mathbb{E}\left[\|X^{\varepsilon,\nu}(s)\mathbbm{1}_{s\leq\tau_{n}}\|^{p}\right]ds$
(16)

Taking $n$ goes to infinity and using Gronwall’s lemma, we prove this bound.

###### Lemma 6

$\left\\{X^{\varepsilon,\nu^{\varepsilon}}\right\\}_{\varepsilon>0}$ is
tightness

###### Proof 3

In view of the Kolmogorov tightness criterion, it suffices to show that there
exist strictly positive constants $\alpha$, $\beta$ and $\gamma$ such that for
all $t$, $s\in[0,T]$,

$\sup_{\nu\in\mathcal{S}_{N}}\mathbb{E}\left[\left|X^{\varepsilon,\nu^{\varepsilon}}(t)-X^{\varepsilon,\nu^{\varepsilon}}(s)\right|^{\alpha}\right]\leq\beta\left|t-s\right|^{\gamma}$

Without loss of generality, let $s<t$. We will write
$b^{n}_{s}:=b_{\varepsilon}(s,X^{\varepsilon,\nu^{\varepsilon}}_{s}\mathbbm{1}_{\left\\{s\leq\tau_{n}\right\\}})$
and
$\sigma^{n}_{s}:=\sigma_{\varepsilon}\left(s,X^{\varepsilon,\nu^{\varepsilon}}_{s}\mathbbm{1}_{\left\\{s\leq\tau_{n}\right\\}}\right)$.

$\displaystyle\mathbb{E}\left[\left|X^{\varepsilon,\nu^{\varepsilon}}(t)-X^{\varepsilon,\nu^{\varepsilon}}(s)\right|^{\alpha}\right]$
$\displaystyle\leq
3^{p-1}(t-s)^{p-1}\mathbb{E}\left\\{\int_{s}^{t}\left|b^{n}_{u}\right|^{p}du\right\\}+3^{p-1}\mathbb{E}\left[\left(\int_{s}^{t}\left|\sigma_{u}^{n}\right|\left|\nu(u)\right|du\right)^{p}\right]$
(17)
$\displaystyle+3^{p-1}\vartheta_{\varepsilon}^{P}\mathbb{E}\left[\left|\int_{s}^{t}\sigma_{u}^{n}dW(u)\right|^{p}\right]$
$\displaystyle\leq
3^{p-1}(t-s)^{p-1}\mathbb{E}\left\\{\int_{s}^{t}\left|b^{n}_{u}\right|^{p}du\right\\}+3^{p-1}N^{\frac{p}{2}}\mathbb{E}\left[\int_{s}^{t}\left|\sigma^{n}_{u}\right|^{p}du\right]$
$\displaystyle+3^{p-1}C_{p}\mathbb{E}\left[\int_{s}^{t}|\sigma^{n}_{u}|^{\frac{p}{2}}du\right]$
$\displaystyle\leq
3^{p-1}(t-s)^{p-1}\mathbb{E}\left\\{\int_{s}^{t}\left|b^{n}_{u}\right|^{p}du\right\\}+3^{p-1}N^{\frac{p}{2}}\mathbb{E}\left[\int_{0}^{t}|\sigma^{n}_{u}|^{\frac{p}{2}}du\right]$
$\displaystyle+3^{p-1}C_{p}\mathbb{E}\left[\int_{s}^{t}|\sigma^{n}_{u}|^{\frac{p}{2}}du\right]$

From the linear growth condition on $b_{\varepsilon}$ and
$\sigma_{\varepsilon}$ we deduce that there exists sufficiently large $\beta$
and let $\alpha=p$, $\gamma=p-1$. Then the hypotheses of Kolmogorov’s
criterion are therefore satisfied.

###### Lemma 7

For any positive $N<\infty$, the set

$K_{N}:=\left\\{\mathcal{G}^{0}\left(\int_{0}^{\cdot}\nu(s)ds,\nu\in\mathcal{S}_{N}\right)\right\\}$

is a compact set in $\mathcal{C}\left([0,T];\mathbb{R}^{n}\right)$

###### Proof 4

We first prove $\mathcal{G}^{0}$ is a continuity map form $\mathcal{S}_{N}$ to
$\mathcal{C}\left([0,T];\mathbb{R}\right)$, then for any positive $N<\infty$,
$\mathcal{S}_{N}$ is compact set in weak topology. Since $\mathcal{G}^{0}$ is
continuity map, we can show $K_{N}$ is a compact set in
$\mathcal{C}\left([0,T];\mathbb{R}\right)$.

Taking $\left\\{\nu^{n}(s)\right\\}\in\mathcal{S}_{N}$,
$\nu^{n}\rightarrow\nu$ weakly, let $\varphi^{n}=\mathcal{G}^{0}(\nu^{n})$,
$\varphi=\mathcal{G}^{0}(\nu)$. Then, for $t\in[0,T]$,

$\displaystyle\varphi^{n}(t)-\varphi(t)$
$\displaystyle=\int_{0}^{t}\left(b(s,\varphi^{n})-b(s,\varphi)\right)ds+\int_{0}^{t}(\sigma(s,\varphi^{n})-\sigma(s,\varphi))\nu^{n}(s)ds$
(18) $\displaystyle+\int_{0}^{t}\sigma(s,\varphi_{s})(\nu^{n}(s)-\nu(s))ds$

Since $\|\nu^{n}\|\leq N$, it follows from (8) that
$R:=\sup_{n\in\mathbb{N}}\|\varphi\|\vee\|\varphi^{n}\|$ is finite. Therefore,
using assumption A.4,

$\displaystyle\sup_{0\leq s\leq t}\left|\varphi^{n}(s)-\varphi(s)\right|$
$\displaystyle\leq L_{R}\int_{0}^{t}\sup_{0\leq u\leq
s}\left|\varphi^{n}(u)-\varphi(u)\right|ds+L_{R}\int_{0}^{t}\sup_{0\leq u\leq
s}\left|\varphi^{n}(u)-\varphi(u)\right|\nu^{n}(s)ds$ (19)
$\displaystyle+\sup_{0\leq u\leq
T}\left|\int_{0}^{u}\sigma(s,\varphi_{s})\left(\nu^{n}(s)-\nu(s)\right)ds\right|$

Let $\Delta^{n}_{\sigma}=\sup_{0\leq u\leq
T}\left|\int_{0}^{u}\sigma(s,\varphi_{s})\left(\nu^{n}(s)-\nu(s)\right)ds\right|$.
By Holder’s inequality and since $\|\nu^{n}\|^{2}\leq N$ for all
$n\in\mathbb{N}$, it follows that

$\sup_{0\leq s\leq t}\left|\varphi^{n}(s)-\varphi(s)\right|\leq
3L_{R}^{2}(t+N)\int_{0}^{t}\sup_{0\leq u\leq
s}\left|\varphi^{n}(u)-\varphi(u)\right|^{2}ds+3(\Delta^{n}_{\sigma})^{2}$

By Gronwall’s lemma, we can deduce that

$\mathcal{G}^{0}(\nu^{n})-\mathcal{G}^{0}(\nu)=\sup_{0\leq t\leq
T}|\varphi^{n}(t)-\varphi(t)|^{2}\leq
3\left(\Delta^{n}_{\sigma}\right)^{2}e^{3L^{2}T(T+N)}$

In order to establish continuity of $\mathcal{G}^{0}$ on $\mathcal{S}_{N}$, it
remains to check that $\Delta^{n}_{\sigma}$ goes to $0$ as
$n\rightarrow\infty$. By A.3, it follows that $\sigma(\cdot,\varphi)\nu^{n}$
converges weakly to $\sigma\left(\cdot,\varphi\right)\nu$ in $L^{2}$.
Moreover, the family
$\left\\{\sigma(\cdot,\varphi)\nu^{n}\right\\}_{n\in\mathbb{N}}$ is bounded in
$L^{2}$ with respect to the $L^{2}-$norm. Hence,

$\int_{0}^{t}\sigma(s,\varphi_{s})\nu^{n}(s)ds\rightarrow\int_{0}^{t}\sigma\left(s,\varphi\right)\nu(s)ds\quad\text{as
}n\rightarrow\infty$

which implies that $\Delta^{n}_{\sigma}\rightarrow 0$ as $n\rightarrow\infty$.

###### Lemma 8

Under A.1-A.6, for every $N<+\infty$ and any family
$\left\\{\nu^{\varepsilon}\right\\}_{\varepsilon>0}\in\mathcal{A}_{N}$
satisfying that $\nu^{\varepsilon}$ converge in distribution as
$\mathcal{S}_{N}-$valued random elements to $\nu$ as $\varepsilon\rightarrow
0$,
$\mathcal{G}^{\varepsilon}\left(W_{\cdot}+\frac{1}{\vartheta_{\varepsilon}}\int_{0}^{\cdot}\nu^{\varepsilon}(s)ds\right)$
converges in distribution to
$\mathcal{G}^{0}\left(\int_{0}^{\cdot}\nu(s)ds\right)$ as
$\varepsilon\rightarrow 0$.

###### Proof 5

By Skorohod representation theorem we can work with almost sure convergence
for the purpose of identifying the limit. We follow the technique in Chiarini
and Fischer [2].

For $t\in[0,T]$, define
$\Phi_{t}:\mathcal{S}_{N}\times\mathcal{C}\left([0,T];R^{n}\right)$ as

$\Phi_{t}(\omega,f):=\left|\omega(t)-x_{0}-\int_{0}^{t}b(s,\omega_{s})ds-\int_{0}^{t}\sigma(s,\omega_{s})f(s)ds\right|\wedge
1$

$\Phi_{t}$ is bounded and we show that it is also continuous. Let
$\omega^{n}\rightarrow\omega$ in $\mathcal{C}\left([0,T];R^{n}\right)$ and
$f^{n}\rightarrow f$ in $\mathcal{S}_{N}$ with respect to the weak topology.
A.2 implies the existence of continuous moduli of continuity $\rho_{b}$ and
$\rho_{\sigma}$ for both coefficients such that
$\left|b(t,\varphi_{t})-b(t,\phi_{t})\right|\leq\rho_{b}\left(\|\varphi-\phi\|\right)$
and
$\left|\sigma(t,\varphi_{t})-\sigma(t,\phi_{t})\right|\leq\rho_{\sigma}\left(\|\varphi-\phi\|\right)$.
Using Holder’s inequality we find that

$\displaystyle\left|\Phi_{t}(\omega^{n},f^{n})-\Phi_{t}(\omega^{n},f)\right|$
$\displaystyle\leq\left|\omega^{n}(t)-\omega(t)\right|+\int_{0}^{t}\left|b(s,\omega^{n}_{s})-b(s,\omega_{s})\right|ds$
(20)
$\displaystyle+\int_{0}^{t}\left|\sigma(s,\omega^{n}_{s})-\sigma(s,\omega_{s})\right||f^{n}(s)|ds+\left|\int_{0}^{t}\sigma(s,\omega_{s})(f^{n}(s)-f(s))ds\right|$
$\displaystyle\leq\|\omega^{n}-\omega\|+T\rho_{b}\left(\|\omega^{n}-\omega\|\right)+\sqrt{NT}\rho_{\sigma}\left(\|\omega^{n}-\omega\|\right)$
$\displaystyle+\|\sigma(\cdot,\omega)\|\left|\int_{0}^{t}\left(f^{n}(s)-f(s)\right)ds\right|$

Since $f^{n}$ tends to $f$ weakly in $L^{2}$ then the last integral converges
to zero as $n$ goes to infinity. Moreover
$\lim\limits_{n\uparrow\infty}\|\omega^{n}-\omega\|=0$, which proves that
$\Phi_{t}$ is continuous, and therefore

$\lim\limits_{n\uparrow\infty}\mathbb{E}\left[\Phi_{t}(X^{n},\nu^{n})\right]=\mathbb{E}\left[\Phi_{t}(X,\nu)\right]$

Define
$b^{R}_{\varepsilon}:\left[0,T\right]\times\mathcal{C}([0,T];\mathbb{R}^{d})$
and
$\sigma^{R}_{\varepsilon}:\left[0,T\right]\times\mathcal{C}([0,T];\mathbb{R}^{d})$
by

$b^{R}_{\varepsilon}(s,\omega_{s})=\left\\{\begin{matrix}b_{\varepsilon}(s,\omega_{s})\quad\text{if}\|\omega\|\leq
R\\\
b_{\varepsilon}(s,\frac{R}{\|\omega\|}\omega_{s})\quad\text{otherwise}\end{matrix}\right.\quad\sigma^{R}_{\varepsilon}(s,\omega_{s})=\left\\{\begin{matrix}\sigma_{\varepsilon}(s,\omega_{s})\quad\text{if}\|\omega\|\leq
R\\\
\sigma_{\varepsilon}(s,\frac{R}{\|\omega\|}\omega_{s})\quad\text{otherwise}\end{matrix}\right.$

It is clear that the function $b^{R}_{\varepsilon}$ and
$\sigma^{R}_{\varepsilon}$ are globally Lipschitz and bounded. By assumption
A.2, $b^{R}_{\varepsilon}\rightarrow b^{R}$ and
$\sigma^{R}_{\varepsilon}\rightarrow\sigma^{R}$ uniformly on
$[0,T]\times\mathcal{C}\left([0,T];\mathbb{R}^{n}\right)$. In analogy with
$\Phi_{t}$, set

$\Phi^{R}_{t}(\omega,f):=\left|\omega(t)-x_{0}-\int_{0}^{t}b^{R}(s,\omega_{s})ds-\int_{0}^{t}\sigma^{R}(s,\omega_{s})f(s)ds\right|\wedge
1$

Consider the family $\left\\{X^{R,\varepsilon,\nu}\right\\}$ of solutions to
the PSDE

$X^{R,\varepsilon,v}(t)=X_{0}^{\varepsilon}+\int_{0}^{t}\left[b^{R}_{\varepsilon}\left(s,X_{s}^{R,\varepsilon,\nu}\right)+\sigma^{R}_{R,\varepsilon}\left(s,X_{s}^{R,\varepsilon,\nu}\right)v(s)\right]\mathrm{d}s+\vartheta_{\varepsilon}\int_{0}^{t}\sigma^{R}_{R,\varepsilon}\left(s,X_{s}^{R,\varepsilon,\nu}\right)\mathrm{d}W(s)$

We will show

$\lim\limits_{\varepsilon\rightarrow
0}\mathbb{E}\left[\Phi_{t}^{R}\left(X^{R,\varepsilon,\nu^{\varepsilon}},\nu\right)\right]=0$
$\displaystyle\mathbb{E}\left[\Phi_{t}^{R}\left(X^{R,\varepsilon,\nu^{\varepsilon}},\nu\right)\right]$
$\displaystyle\leq\left|X^{\varepsilon}_{0}-X_{0}\right|+\mathbb{E}\left[\int_{0}^{t}\left|b^{R}_{\varepsilon}(s,X^{R,\varepsilon,\nu^{\varepsilon}})-b^{R}(s,X^{R,\varepsilon,\nu^{\varepsilon}})\right|ds\right]$
(21)
$\displaystyle+\mathbb{E}\left[\int_{0}^{t}\left|\sigma^{R}_{\varepsilon}(s,X^{R,\varepsilon,\nu^{\varepsilon}})-\sigma^{R}(s,X^{R,\varepsilon,\nu^{\varepsilon}})\right||\nu(s)|ds\right]$
$\displaystyle+\vartheta_{\varepsilon}\mathbb{E}\left[\left|\int_{0}^{t}\sigma^{R}_{\varepsilon}\left(s,X^{R,\varepsilon,\nu^{\varepsilon}}\right)dW(s)\right|\right]$
$\displaystyle\leq\left|X^{\varepsilon}_{0}-X_{0}\right|+t\|b^{R}_{\varepsilon}-b^{R}\|+\|\sigma^{R}_{\varepsilon}-\sigma^{R}\|\mathbb{E}\left[\int_{0}^{T}|\nu(s)|ds\right]$
$\displaystyle+\vartheta_{\varepsilon}\sqrt{\int_{0}^{t}\mathbb{E}\left[\sigma_{\varepsilon}^{R}(s,X^{R,\varepsilon,\nu^{\varepsilon}})^{2}\right]ds}$

The last term in the above display tends to 0 since

$\displaystyle\sup_{\nu\in\mathcal{S}_{N}}\int_{0}^{t}\mathbb{E}\left[\sigma_{\varepsilon}^{R}(s,X^{R,\varepsilon,\nu^{\varepsilon}})^{2}\right]ds$
$\displaystyle\leq
2\sup_{\nu\in\mathcal{S}_{N}}\int_{0}^{T}\mathbb{E}\left[\left|\sigma^{R}(s,X^{R,\varepsilon,\nu^{\varepsilon}})\right|^{2}\right]ds$
(22)
$\displaystyle+2\sup_{\nu\in\mathcal{S}_{N}}\int_{0}^{T}\mathbb{E}\left[\left|\sigma^{R}_{\varepsilon}(s,X^{R,\varepsilon,\nu^{\varepsilon}})-\sigma^{R}(s,X^{R,\varepsilon,\nu^{\varepsilon}})\right|^{2}\right]ds$
$\displaystyle\leq
2T\sup_{\nu\in\mathcal{S}_{N}}\|\sigma_{\varepsilon}^{R}-\sigma^{R}\|^{2}+2\sup_{\nu\in\mathcal{S}_{N}}\int_{0}^{T}\mathbb{E}\left[\left|\sigma^{R}(S,X^{R,\varepsilon,\nu^{\varepsilon}})\right|^{2}\right]ds$
$\displaystyle<\infty$

Then, we have

$\lim\limits_{n\rightarrow\infty}\mathbb{E}\left[\Phi_{t}^{R}\left(X^{R,\varepsilon,\nu^{\varepsilon}},\nu\right)\right]=0$

For $R>0$ and $\nu\in\mathcal{S}_{N}$, let $\tau^{R}_{n}$ is a stopping time
defined by

$\tau^{R}_{n}=\inf\left\\{t\geq 0:X^{\varepsilon,\nu}(t)\geq R\right\\}$

We have

$\mathbb{P}\left(X^{R,\varepsilon,\nu^{\varepsilon}}(t)=X^{\varepsilon,\nu^{\varepsilon}}(t)\mathbbm{1}_{t\leq\tau_{n}}\right)=1$

It follows that

$\displaystyle\mathbb{E}\left[\Phi_{t}(X^{\varepsilon,\nu},\nu)\right]$
$\displaystyle=\mathbb{E}\left[\mathbbm{1}_{t<\tau_{n}}\Phi_{t}(X^{\varepsilon,\nu},\nu)\right]+\mathbb{E}\left[\mathbbm{1}_{t\geq\tau_{n}}\Phi_{t}(X^{\varepsilon,\nu},\nu)\right]$
(23)
$\displaystyle\leq\mathbb{E}\left[\Phi^{R}_{t}(X^{R,\varepsilon,\nu^{\varepsilon}},\nu)\right]+\mathbb{P}\left(t\geq\tau_{R}^{n}\right)$

For all $\nu\in\mathcal{S}_{N}$, by Markov’s inequality we have

$\mathbb{P}\left(t\geq\tau^{R}_{n}\right)=\mathbb{P}\left(\sup_{0\leq s\leq
t}|X^{R,\varepsilon,\nu^{\varepsilon}}(s)|\geq R\right)\leq\frac{c}{R^{2}}$

Taking upper limits on both side of (23), we obtain

$\limsup_{\varepsilon\rightarrow
0}\mathbb{E}\left[\Phi_{t}(X^{\varepsilon,\nu},\nu)\right]\leq\limsup_{n\rightarrow\infty}\mathbb{P}\left(t\geq\tau^{R}_{n}\right)\leq\frac{c}{R}$

Since $R>0$ has been chosen arbitrarily, it follows that

$\lim\limits_{\varepsilon\rightarrow
0}\mathbb{E}\left[\Phi_{t}(X^{\varepsilon,\nu^{\varepsilon}},\nu^{\varepsilon})\right]=0$

###### Proof of Theorem 3 1

According to Theorem 2, combined with Lemma 5, 6, 7 and 8 , it can be seen
that Theorem 3 holds.

##  4 Application: Small time large deviation principle for path-dependent
stochastic differential equation

In this section, we study the LDP for functional of PSDEs in small time
intervals: $\left\\{X(t),t\in\mathbb{T}\right\\}$ as $t\rightarrow 0$, where

$X(t)=x_{0}+\int_{0}^{t}b(s,X_{s})ds+\int_{0}^{t}\sigma(s,X_{s})dW(s)$

We rescale the small time problem to a small perturbation problem.

$\displaystyle X({\varepsilon t})$ $\displaystyle=x_{0}+\int_{0}^{\varepsilon
t}b(s,X_{s})ds+\int_{0}^{\varepsilon t}\sigma(s,X_{s})dW(s)$ (24)
$\displaystyle=x_{0}+\varepsilon\int_{0}^{t}b(\varepsilon t,X_{\varepsilon
t})ds+\sqrt{\varepsilon}\int_{0}^{\varepsilon t}\sigma(\varepsilon
t,X_{\varepsilon t})d\widehat{W}(s)$

Where, $\widehat{W}(s)=\frac{1}{\sqrt{\varepsilon}}W(\varepsilon s)$. Let
$U(t)=X(\varepsilon t)$, by (24), we have

$U(t)=x_{0}=\int_{0}^{t}b_{\varepsilon}(s,U_{s})ds+\sqrt{\varepsilon}\int_{0}^{t}\sigma(s,U_{s})d\widehat{W}(s)$

Now, we can use Theorem 3 to obtain the LDP for small time PSDEs.

###### Theorem 9

The process $X(\varepsilon t)$ satisfies LDP as $\varepsilon\rightarrow 0$
with rate function $J^{1}$ and speed $\varepsilon$.

$J(g)=\inf_{\nu\in
L^{2};g=\mathcal{G}^{0}(\int_{0}^{\cdot}\nu(s)ds)}\left\\{\frac{1}{2}\int_{0}^{\cdot}|\nu(s)|^{2}ds\right\\}$

$\mathcal{G}^{0}$ is the solution map of (25)

$\phi(t)=x_{0}+\int_{0}^{t}\sigma(s,\phi_{s})ds$ (25)

For functional of $X(\varepsilon t)$ generally, we have the following result

###### Theorem 10

Let $f\in\mathcal{C}^{1}_{b}(\mathbb{R}^{d};\mathbb{R}^{m})$. Then the process
$f(X(\varepsilon t))$ satisfies a LDP as $\varepsilon\rightarrow 0$ with rate
function $J^{f}$ and speed $\varepsilon$.

$J^{f}(g)=\inf_{\left\\{Df(x_{0})\varphi=g\right\\}}{J(\varphi)}$ (26)

###### Proof 6

This proof bases on Theorem 3 and the delta method of large deviation[10].

For any $f\in C_{b}^{1}\left(\mathbb{R}^{d};\mathbb{R}^{m}\right)$, define
$\Phi:C_{0}^{1}\left([0,T],\mathbb{R}^{d}\right)\rightarrow
C_{0}^{1}\left([0,T],\mathbb{R}^{m}\right)$ as follows:

$f(\varphi)(t)=f(\varphi(t)),\quad t\in[0,T].$

$\Phi$ is Hadamard differentiable and its Hadamard differential at constant
function $\varphi\equiv$ $f\left(x_{0}\right)$ is

$\Phi_{f\left(x_{0}\right)}^{\prime}(\psi)=(Df)\left(x_{0}\right)\psi,\quad\psi\in
C_{0}^{\alpha}\left([0,T],\mathbb{R}^{d}\right)$

Then the result follows from the delta method.

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# Positive definite functions on a regular domain

Martin Buhmann  Justus-Liebig University, Lehrstuhl Numerische Mathematik,
35392 Giessen, Germany<EMAIL_ADDRESS>and  Yuan Xu
Department of Mathematics, University of Oregon, Eugene, OR 97403–1222, USA
<EMAIL_ADDRESS>
###### Abstract.

We define positive and strictly positive definite functions on a domain and
study these functions on a list of regular domains. The list includes the unit
ball, conic surface, hyperbolic surface, solid hyperboloid, and simplex. Each
of these domains is embedded in a quadrant or a union of quadrants of the unit
sphere by a distance preserving map, from which characterizations of positive
definite and strictly positive definite functions are derived for these
regular domains.

###### Key words and phrases:

Positive definite function, strictly positive definite function, unit sphere,
unit ball, conic surface, hyperbolic domains, simplex.

###### 2010 Mathematics Subject Classification:

33C45, 41A05, 41A63, 43A90

The second author thanks the Alexander von Humboldt Foundation for an AvH
award that supports his visit to U. Giessen, during which the work was carried
out; he was partially supported by Simons Foundation Grant #849676.

##  1\. Introduction

A central topic in the approximation of functions in several variables is the
approximation by interpolation. Given real-valued data, call them $b_{i}$,
$i=1,2,\ldots,N$, and a set of distinct points
$\\{x_{1},x_{2},\ldots,x_{N}\\}\subset\Omega\subset{\mathbb{R}}^{d}$, the task
of finding an interpolant from a given linear space, e.g., spanned by shifts
or otherwise of a kernel-function, is to find $g:\Omega\mapsto{\mathbb{R}}$
with

$g(x_{i})=b_{i},\qquad\forall\;1\leq i\leq N.$

This $g$ can for instance be formed as a sum of univariate basis functions
$\varphi$, say, combined with a distance function
${\mathsf{d}}_{\Omega}:\Omega\times\Omega\mapsto{\mathbb{R}}$: Then it reads

$g(x)=\sum_{i=1}^{N}\lambda_{i}\varphi({\mathsf{d}}_{\Omega}(x,x_{i})),\qquad
x\in\Omega.$

Therefore, the interpolation problem is uniquely solvable as soon as the
interpolation matrix
$[\varphi({\mathsf{d}}_{\Omega}(x_{i},x_{j}))]_{i,j=1}^{N}$ is non-singular.
Clearly, if $\varphi$ is such that this matrix is positive definite, the said
condition is fulfilled, but other choices are possible (see, e.g., [7]). If no
other properties of $\Omega$ are required other than that it be a subset of
${\mathbb{R}}^{d}$, the Euclidean distance function will be an obvious choice
for the distance function, and then the whole construction comes down to
interpolation with the so-called radial basis functions ([4]).

Interpolation on spheres and other regular domains is a special case of the
great theoretical and practical interest of the above general Ansatz and the
choice of the distance function becomes particularly critical. For the unit
sphere $\SS^{d}$, the distance function is the geodesic distance

${\mathsf{d}}_{\SS}(\xi,\eta)=\arccos{\langle}\xi,\eta{\rangle},\qquad\xi,\eta\in\SS^{d},$

and PDFs on the unit sphere have been studied extensively. The aim of the
present paper is to study positive definite functions on a bounded regular
domain, more general than the unit sphere.

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^{d}$ and let
${\mathsf{d}}_{\Omega}(\cdot,\cdot)$ be a distance function on $\Omega$. We
assume that the domain is dilated so that ${\mathsf{d}}_{\Omega}(x,y)\leq\pi$
for all $x,y\in\Omega$. Let $f:[-1,1]\mapsto{\mathbb{R}}$ be a continuous
function. For $N\in{\mathbb{N}}$, let

$\Xi_{N}=\\{x_{1},\ldots,x_{N}\\}\subset\Omega$

be a set of distinct points in $\Omega$. We denote by $f[\Xi_{N}]$ the
$N\times N$ matrix

$f[\Xi_{N}]=\left[f(\cos{\mathsf{d}}_{\Omega}(x_{i},x_{j}))\right]_{i,j=1}^{N}.$

In other words, $\varphi$ from the above becomes $f(\cos(\cdot))$.

######  Definition 1.1.

A continuous function $f:[-1,1]\mapsto{\mathbb{R}}$ is called a positive
definite function (PDF) on $\Omega$ if for every $N\in{\mathbb{N}}$ and set of
$N$ distinct points $\Xi_{N}=\\{x_{1},\ldots,x_{N}\\}\subset\Omega$, the
$N\times N$ matrix $f[\Xi_{N}]$ is nonnegative definite, and it is called a
strictly positive definite function (SPDF) if $f[\Xi_{N}]$ is positive
definite.

When $\Omega$ is the unit sphere, PDFs are first studied in the seminal work
[18], where the space of PDFs is characterized as the function $f$ whose
Fourier-Gegenbauer coefficients $\hat{f}_{n}$ are all nonnegative. Motivated
by the problem of scatted data interpolation, SPDFs on the unit sphere were
first considered in [23], which gives as a sufficient condition that all
$\hat{f}_{n}>0$, and they are later characterized in [9] as the function $f$
that has infinitely many even and infinitely many odd positive $\hat{f}_{n}$.
These functions are important in several topics in approximation theory,
computational analysis, probability, and statistics. They have generated
considerable interest and led to numerous studies and generalizations. We
refer to [2, 3, 5, 6, 8, 13, 14, 15, 17] for several recent studies that
generalize the concept of PDFs and SPDFs in several different directions.
There are many more papers on PDF functions and their applications. We refer
to [12] for a survey that includes discussions on applications in probability
and other fields and to [16] for some applications in computational analysis.

In the present paper, we consider regular domains $\Omega$ that include the
unit ball, conic surface, hyperbolic surface, solid hyperboloid, and simplex,
on which PDFs are defined via their respective distance functions.

It turns out that each of these domains can be mapped to a portion of the unit
sphere via a distance preserving map, which leads us to consider PDFs on
quadrants of the unit spheres, such as the positive quadrant or upper
hemisphere. This provides a large class of PDFs on the regular domain. Our
main result provides a characterization of SPDFs on these quadrants and each
of the regular domains in our list among this class of PDFs.

One of the essential tools for studying PDFs on the unit sphere is the
addition formula of spherical harmonics, which gives a closed-form formula for
the reproducing kernel of the space of spherical harmonics of a fixed degree.
For a regular domain, the analog of the addition formula is a closed-form
formula for the reproducing kernel of the space of orthogonal polynomials of
fixed degree, where the orthogonality is defined in $L^{2}(\Omega,w)$, with
$w$ being an analogy of the Chebyshev weight function on the domain. For each
of our regular domains, an analog of the addition formula is explicitly given,
which turns out to be an average of a PDF function over reflections of a base
point. The latter resembles the image method for finding the Green’s function
for the Poisson equation over a quadrant in ${\mathbb{R}}^{d}$.

The paper is organized as follows. The next section is preliminary, where we
review background materials and the results on the unit sphere. We discuss
characterizations of PDFs and SPDFs on quadrants of the unit sphere in the
third section and on our list of regular domains in the fourth section.

##  2\. Preliminaries

We review the results on the unit sphere in the first subsection, the
reproducing kernels of orthogonal polynomials in the second subsection, and
state a couple of simple properties of PDFs in the third subsection.

###  2.1. PDFs on the unit sphere

The characterization of PDFs on the unit sphere relies on the Fourier-
Gegenbauer expansions, which we review first. For ${\lambda}>-\frac{1}{2}$,
let the weight function be

$w_{\lambda}(t)=(1-t^{2})^{{\lambda}-\frac{1}{2}},\quad-1\leq t\leq 1.$

The Gegenbauer polynomials $C_{n}^{\lambda}$ are orthogonal polynomials of
degree $n$ that satisfy

$c_{\lambda}\int_{-1}^{1}C_{n}^{\lambda}(t)C_{m}^{\lambda}(t)w_{\lambda}(t)\,\mathrm{d}t=h_{n}^{\lambda}\delta_{m,n},\qquad
h_{n}^{\lambda}=\frac{{\lambda}}{n+{\lambda}}C_{n}^{\lambda}(1),$

where
$c_{\lambda}=\Gamma({\lambda}+\frac{1}{2})\Gamma(\frac{1}{2})/\Gamma({\lambda})$
and $C_{n}^{\lambda}(1)=(2{\lambda})_{n}/n!$. The expression $(\cdot)_{n}$
denotes the Pochhammer symbol. The Fourier-Gegenbauer series of $f\in
L^{2}([-1,1],w_{\lambda})$ is defined by

(2.1)
$f(\cos{\theta})=\sum_{n=0}^{\infty}\hat{f}_{n}^{\lambda}C_{n}^{\lambda}(\cos{\theta}),\quad{\lambda}>-\tfrac{1}{2},$

where the coefficients $\hat{f}_{n}^{\lambda}$ are given by

$\hat{f}_{n}^{\lambda}=[h_{n}^{\lambda}]^{-\frac{1}{2}}c_{\lambda}\int_{0}^{\pi}f(\cos{\theta})C_{n}^{\lambda}(\cos{\theta})(\sin{\theta})^{2{\lambda}}\mathrm{d}{\theta}.$

We can now state the characterization of PDFs on the unit sphere in [18].

######  Theorem 2.1.

Let $d\geq 2$ and ${\lambda}=\frac{d-1}{2}$. A continuous function
$f:[-1,1]\mapsto{\mathbb{R}}$ is positive definite on $\SS^{d}$ if and only if
$\hat{f}_{n}^{\lambda}\geq 0$ for all $n$, in which case the series (2.1)
converges absolutely and uniformly to $f(\cos{\theta})$ on $[0,\pi]$. The most
general $f$ that is positive definite on $\SS^{d}$ is therefore given by the
expansion

(2.2)
$f(\cos{\theta})=\sum_{n=0}^{\infty}a_{n}C_{n}^{\frac{d-1}{2}}(\cos{\theta}),\qquad
a_{n}\geq 0,\quad\forall n\in{\mathbb{N}}_{0}.$

For the strictly positive definite functions, we only need to consider $f$
that has nonnegative Fourier-Gegenbauer coefficients since
$\mathrm{S}\Phi(\SS^{d})\subset\Phi(\SS^{d})$. For such $f$, we can write
(2.1) as

(2.3)
$f=\sum_{n=0}^{\infty}\hat{f}_{n}^{\lambda}C_{n}^{\lambda}=\sum_{n\in{\mathcal{F}}}\hat{f}_{n}^{\lambda}C_{n}^{\lambda},$

where ${\mathcal{F}}$ is the index set defined by

(2.4) ${\mathcal{F}}:=\\{n\in{\mathbb{N}}_{0}:\hat{f}_{n}>0\\}.$

The characterization of PDFs on the unit sphere in [9] is stated as follows.

######  Theorem 2.2.

Let $d\geq 2$ and ${\lambda}=\frac{d-1}{2}$. A continuous function
$f:[-1,1]\mapsto{\mathbb{R}}$ is strictly positive definite if and only if
${\mathcal{F}}$ contains infinitely even integers and infinitely odd integers.

The proofs of these theorems rely on the properties of spherical harmonics and
the Gegenbauer polynomials, which will also be needed in the next section when
we consider PDFs on quadrants of the unit sphere. We recall the necessary
definitions below.

When ${\lambda}=\frac{d-1}{2}$, the Gegenbauer polynomials are often called
ultraspherical polynomials since they are associated with spherical harmonics.
Let ${\mathcal{H}}_{n}^{d+1}$ be the space of spherical harmonics of degree
$n$, which are homogeneous polynomials in $d+1$ variables restricted to the
sphere $\SS^{d}$. Let
$\\{Y_{\ell}^{n}:1\leq\ell\leq\dim{\mathcal{H}}_{n}^{d+1}\\}$ be an
orthonormal basis of ${\mathcal{H}}_{n}^{d}$. Then

$\frac{1}{{\omega}_{d}}\int_{\SS^{d}}Y_{\ell}^{n}(\xi)Y_{\ell^{\prime}}^{m}(\xi)\,\mathrm{d}\sigma(\xi)=\delta_{\ell,\ell^{\prime}}\delta_{n,m},$

where $\mathrm{d}\sigma$ is the surface measure of $\SS^{d}$ and
${\omega}_{d}$ denotes the surface area of $\SS^{d}$. An orthonormal basis of
${\mathcal{H}}_{n}^{d+1}$ can be found explicitly by using spherical
coordinates; see, for example, [11, p. 116]. The reproducing kernel of the
space ${\mathcal{H}}_{n}^{d+1}$ is a function ${\mathsf{P}}_{n}(\cdot,\cdot)$
such that

$\frac{1}{{\omega}_{d}}\int_{\SS^{d}}{\mathsf{P}}_{n}(\xi,\eta)Q(\eta)\,\mathrm{d}\sigma(\eta)=Q(\xi),\qquad\forall
Q\in{\mathcal{H}}_{n}^{d}.$

Let $\\{Y_{\ell}^{n}:1\leq\ell\leq\dim{\mathcal{H}}_{n}^{d}\\}$ be an
orthonormal basis of ${\mathcal{H}}_{n}^{d}$. One of the most important
properties of the spherical harmonics is their addition formula, which gives a
closed-form formula for the reproducing kernel (cf. [10, (1.2.8)])

(2.5)
${\mathsf{P}}_{n}(\xi,\eta)=\sum_{\ell=1}^{\dim{\mathcal{H}}_{n}^{d+1}}Y_{\ell}^{n}(\xi)Y_{\ell}^{n}(\eta)=Z_{n}^{\frac{d-1}{2}}({\langle}\xi,\eta{\rangle}),\quad\forall\xi,\eta\in\SS^{d},$

where $Z_{n}^{\lambda}$ is a constant multiple of the Gegenbauer polynomial,

(2.6)
$Z_{n}^{\lambda}(t):=\frac{n+{\lambda}}{{\lambda}}C_{n}^{\lambda}(t),\qquad{\lambda}\geq-\tfrac{1}{2}.$

We use the notation ${\langle}\cdot,\cdot{\rangle}$ for inner products here.

As an immediate consequence of (2.5), for example, it leads to

${\mathbf{c}}^{\mathsf{T}}C_{n}^{\frac{d-1}{2}}[\Xi_{N}]{\mathbf{c}}=\frac{{\lambda}}{n+{\lambda}}\sum_{\ell=1}^{\dim{\mathcal{H}}_{n}^{d+1}}\left[\sum_{k=1}^{N}c_{\ell}Y_{\ell}^{n}(\xi_{k})\right]^{2}\geq
0$

for any ${\mathbf{c}}\in{\mathbb{R}}^{N}$, which shows that
$C_{n}^{\frac{d-1}{2}}\in\Phi(\SS^{d})$ for each $n\geq 0$. Here, we use the
notation $\xi_{k}$ instead of $x_{k}$ for our elements of the set $\Xi_{N}$ as
long as we are on the sphere.

Writing in terms of the orthonormal basis derived via the separation of
variables in spherical coordinates, (2.5) leads to the addition formula for
the Gegenbauer polynomials. For ${\lambda}>0$, the latter states [1, (9.85)]

(2.7)
$C_{n}^{\lambda}(\cos{\theta}\cos\phi+u\sin{\theta}\sin\phi)=\sum_{k=0}^{n}b_{k,n}^{\lambda}Q_{k,n}^{\lambda}({\theta})Q_{k,n}^{\lambda}(\phi)C_{k}^{{\lambda}-\frac{1}{2}}(u),$

where the coeffients $b_{k,n}^{\lambda}$ are positive constants and

$Q_{k,n}^{\lambda}({\theta})=(\sin{\theta})^{k}C_{n-k}^{{\lambda}+k}(\cos{\theta}),\quad
0\leq k\leq n.$

###  2.2. Reproducing kernels for orthogonal polynomials

Let $w$ be a non-negative weight function defined on the compact domain
$\Omega\subset{\mathbb{R}}^{d}$. Then the bilinear form with weight

${\langle}f,g{\rangle}_{w}=\int_{\Omega}f(x)g(x)w(x)\,\mathrm{d}x$

defines an inner product of $L^{2}(\Omega,w)$. We consider orthogonal
polynomials for this inner product, which are well defined among polynomials
restricted to $\Omega$. Notice that when $\Omega$ is a quadratic surface
defined by $\phi(x)=0$, such as the unit sphere, the polynomials restricted on
$\Omega$ belong to the polynomial ring
${\mathbb{R}}[x]/{\langle}\phi{\rangle}$.

Let ${\mathcal{V}}_{n}(\Omega)$ be the space of orthogonal polynomials of
degree at most $n$.

######  Definition 2.3.

A function $P_{n}(w;\cdot,\cdot)$ is called a reproducing kernel of
${\mathcal{V}}_{n}(\Omega)$ if

  1. (1)

$P_{n}(w;\cdot,\cdot)$ is symmetric: $P_{n}(w;x,y)=P_{n}(w;y,x)$;

  2. (2)

$P_{n}(w;\cdot,y)\in{\mathcal{V}}_{n}(\Omega)$ for all $y\in\Omega$;

  3. (3)

$P_{n}(w;\cdot,\cdot)$ reproduces ${\mathcal{V}}_{n}(\Omega)$ in the sense
that

(2.8) $\int_{\Omega}P_{n}(w;x,y)Q(y)w(y)\,\mathrm{d}y=Q(x),\qquad
Q\in{\mathcal{V}}_{n}(\Omega).$

The kernel is uniquely determined by these properties. Furthermore, if
$\\{P_{\ell}^{n}\\}_{\ell=1}^{a(n,d)}$ is an orthonormal basis of
${\mathcal{V}}_{n}(\Omega)$, then

$P_{n}(w;x,y)=\sum_{k=1}^{a(n,d)}P_{k}^{n}(x)P_{k}^{n}(y),\qquad
a(n,d)=\dim{\mathcal{V}}_{n}(\Omega).$

We refer to [11, Section 3.6] for further properties. The reproducing kernels
are important for understanding the Fourier orthogonal expansion. Indeed, for
$f\in L^{2}(\Omega,w)$, its Fourier orthogonal series is defined by

$f=\sum_{n=0}^{\infty}\sum_{\ell=1}^{a(n,d)}\widehat{f}_{\ell,n}P_{\ell}^{n},\quad\hbox{where}\quad\widehat{f}_{\ell,n}={\langle}f,P_{\ell}^{n}{\rangle}_{w}.$

Then $P_{n}(w;\cdot,\cdot)$ is the kernel of the projection operator
$\operatorname{proj}_{n}:L^{2}(\Omega,2)\mapsto{\mathcal{V}}_{n}(\Omega)$,

$\operatorname{proj}_{n}f(x)=\sum_{\ell=1}^{a(n,d)}\widehat{f}_{\ell,n}P_{\ell}^{n}(x)=\int_{\Omega}f(y)P_{n}(w;x,y)w(y)\,\mathrm{d}y,\qquad
x\in\Omega.$

For the analysis of the Fourier orthogonal expansion, a closed-form formula of
the reproducing kernel $P_{n}(w;\cdot,c\dot{)}$ is often essential. The
addition formula (2.5) for the unit sphere provides one of the most elegant
formulas in this regard. We will encounter similar closed-form formulas on our
regular domains and will refer to them as addition formulas as well.

Finally, a word about our notation. Namely, we shall use different fonts for
the kernels on different types of domains: ${\mathsf{P}}_{n}$ if $\Omega$ is a
hypersurface and ${\mathbf{P}}_{n}$ if $\Omega$ is a solid domain.

###  2.3. PDFs on a domain

For convenience, we denote $\Phi(\Omega)$ the space of all positive definite
functions on $\Omega$ and by $\mathrm{S}\Phi(\Omega)$ the space of all
strictly positive definite functions on $\Omega$ throughout the rest of the
paper.

Several properties for $\Phi(\Omega)$ follow readily from the theory of
matrices.

  1. (1)

If $f_{j}\in\Phi(\Omega)$ and $c_{j}\geq 0$ for $1\leq j\leq n$, then
$f=c_{1}f_{1}+\cdots+c_{n}f_{n}\in\Phi(\Omega)$.

  2. (2)

If $f\in\Phi(\Omega)$ and $g\in\Phi(\Omega)$, then $fg\in\Phi(\Omega)$,

and both properties hold for $\mathrm{S}\Phi(\Omega)$. While the first one is
evident from the definition of PDFs, the second one follows from Schur’s
product theorem, which states that the Hadamard product (or entrywise product)
of two positive definite matrices is also a positive definite matrix.

##  3\. Positive definite functions on quadrants of the unit sphere

In this section, we study PDFs and SPDFs on quadrants of the unit sphere. For
a positive integer $k$, $1\leq k\leq d$, we define $\SS^{d}_{k,+}$ by

$\SS_{k,+}^{d}=\left\\{\xi\in\SS^{d}:\xi_{k}\geq 0,\ldots,\xi_{d+1}\geq
0\right\\}.$

In particular, $\SS_{d+1,+}^{d}$ is the upper hemisphere of $\SS^{d}$ and
$\SS_{1,+}^{d}$ is the positive quadrant of $\SS^{d}$. We consider
$\Phi(\SS^{d}_{k,+})$ with $\SS_{k,+}^{d}$ equipped with the geodesic distance
${\mathsf{d}}_{\SS^{d}_{k,+}}(\cdot,\cdot)={\mathsf{d}}_{\SS}(\cdot,\cdot)$
inherited from $\SS^{d}$. In particular, it follows that
$\Phi(\SS^{d})\subset\Phi(\SS^{d}_{k,+})$.

The class of PDFs for $\SS^{d}_{k,+}$ is again given in terms of the Fourier-
Gegenbauer series (2.1) and is similar to that of Theorem 2.1. We shall not
include the convergence of the series in the statement in the rest of the
paper.

######  Theorem 3.1.

Let $d\geq 2$ and ${\lambda}=\frac{d-1}{2}$. Let $k$ be an integer such that
$1\leq k\leq d+1$. A continuous function $f:[-1,1]\mapsto{\mathbb{R}}$ is
positive definite on $\SS_{k,+}^{d}$ if $\hat{f}_{n}^{\lambda}\geq 0$ for all
$n$. Moreover, if $k=d+1$, then it is also necessary; in particularu,
$\Phi(\SS_{d+1,+})=\Phi(\SS^{d})$.

###### Proof.

Since $\Phi(\SS^{d})\subset\Phi(\SS^{d}_{k,+})$, the sufficient direction
follows immediately from the classical result for $\Phi(\SS^{d})$. In
particular, $C_{n}^{{\lambda}}\in\Phi(\SS^{d}_{k,+})$ for all $n\geq 0$. We
now prove the other direction for $k=d+1$. Assume that $f$ is a PDF on
$\SS_{d+1,+}^{d}$, so we need to show that $\hat{f}_{n}^{\lambda}\geq 0$. For
$\Phi(\SS^{d})$, this is established by using the identiy

$\displaystyle\int_{\SS^{d}}f({\langle}\xi,\eta{\rangle})\,\mathrm{d}\sigma(\xi)=\frac{1}{{\omega}_{d}}\int_{\SS^{d}}\int_{\SS^{d}}f({\langle}\xi,\eta{\rangle})\,\mathrm{d}\sigma(\eta)\,\mathrm{d}\sigma(\xi),$

where ${\omega}_{d}$ denotes the surface area of $\SS^{d}$. For $1\leq k\leq
d+1$ and $\xi\in\SS^{d}$, we define
$\xi_{k,+}:=(\xi_{1},\ldots,\xi_{k-1},|\xi_{k}|,\ldots,|\xi_{d+1}|)\in\SS_{k,+}^{d}$
and similarly for $\eta_{k,+}$. The above identity is proved by using the
invariant of $\mathrm{d}\sigma$ on $\SS^{d}$, which holds if $\eta$ is
replaced by $\eta_{k,+}$, since there is an rotation $h$ such that
$h\eta_{k,+}=e$, so that

$\displaystyle\int_{\SS^{d}}\int_{\SS^{d}}f({\langle}\xi,\eta_{k,+}{\rangle})\,\mathrm{d}\sigma(\eta)\mathrm{d}\sigma(\xi)$
$\displaystyle=\int_{\SS^{d}}\int_{\SS^{d}}f({\langle}h\xi,e{\rangle})d\sigma(\eta)\mathrm{d}\sigma(\xi)$
$\displaystyle={\omega}_{d}\int_{\SS^{d}}f({\langle}\xi,e{\rangle})\,\mathrm{d}\sigma(\xi)={\omega}_{d}\int_{\SS^{d}}f({\langle}\xi,\eta_{k,+}{\rangle})\,\mathrm{d}\sigma(\xi),$

where the second step follows from rotation invariance of $\mathrm{d}\sigma$.
We claim that if $f$ is even and PDF, then

(3.1)
${\omega}_{d}\int_{\SS^{d}}f({\langle}\xi,e{\rangle})\mathrm{d}\sigma(\xi)=4\int_{\SS_{d+1,+}^{d}}\int_{\SS_{d+1,+}^{d}}f({\langle}\xi,\eta{\rangle})\,\mathrm{d}\sigma(\eta)\mathrm{d}\sigma(\xi)\geq
0.$

For simplicity, we write $\SS_{+}^{d}=\SS_{d+1,+}^{d}$ in the proof below. Let
$\SS_{-}^{d}=\\{\xi\in\SS^{d}:\xi_{d+1}\leq 0\\}$ and
$\xi_{d+1,-}=(\xi^{\prime},-\xi_{d+1})$ for $\xi\in\SS^{d}$. Then

$\displaystyle\int_{\SS^{d}}\int_{\SS_{-}^{d}}f({\langle}\xi,\eta_{d+1,+}{\rangle})\mathrm{d}\sigma(\xi)\mathrm{d}\sigma(\eta)$
$\displaystyle=\int_{\SS^{d}}\int_{\SS_{+}^{d}}f({\langle}\xi_{d+1,-},\eta_{d+1,+}{\rangle})\mathrm{d}\sigma(\xi)\mathrm{d}\sigma(\eta)$
$\displaystyle=\int_{\SS^{d}}\int_{\SS_{+}^{d}}f(-{\langle}\xi,\eta_{d+1,+}{\rangle})\,\mathrm{d}\sigma(\xi)\mathrm{d}\sigma(\eta)$
$\displaystyle=\int_{\SS^{d}}\int_{\SS_{+}^{d}}f({\langle}\xi,\eta_{d+1,+}{\rangle})\,\mathrm{d}\sigma(\xi)\mathrm{d}\sigma(\eta),$

where the second identity follows from changing variables
$\eta_{j}\mapsto-\eta_{j}$ for $1\leq j\leq d$ and the last identity uses $f$
being even. Putting these integrals together, we have proved

$\displaystyle\int_{\SS^{d}}\int_{\SS^{d}}f({\langle}\xi,\eta_{k,+}{\rangle})\,\mathrm{d}\sigma(\eta)\mathrm{d}\sigma(\xi)=2\int_{\SS^{d}}\int_{\SS_{+}^{d}}f({\langle}\xi,\eta_{d+1,+}{\rangle})\,\mathrm{d}\sigma(\xi)\mathrm{d}\sigma(\eta),$

from which the identity of (3.1) follows readily, sicne $\eta_{d+1,+}$ is even
in its $d+1$ variable. Since $f\in\Phi(\SS_{d+1,+}^{d})$, applying a positive
cubature rule of degree $n$ on $\SS_{d+1,+}^{d}$ shows that the right-hand
side is nonnegative for all polynomials of degree $\leq n$ and, by the
Weierstraß theorem about the density of polynomials, for all continuous
functions. Hence, we have proved that the integral in (3.1) is non-negative.

Now, for $d\geq 2$, a change of variable shows that

$\int_{\SS^{d}}f({\langle}\xi,\eta{\rangle})\,\mathrm{d}\sigma(\eta)={\omega}_{d-1}\int_{0}^{\pi}f(\cos{\theta})(\sin{\theta})^{d-1}\,\mathrm{d}{\theta}.$

Hence, setting $e=e_{d+1}=(0,\ldots,0,1)\in{\mathbb{R}}^{d+1}$ as a
$(d+1)$-dimensional vector and using the invariance of $\mathrm{d}\sigma$, we
see that

$\hat{f}_{n}^{\lambda}=[h_{n}^{\lambda}]^{-\frac{1}{2}}\frac{1}{{\omega}_{d}}\int_{\SS^{d}}f({\langle}e,\xi{\rangle})C_{n}^{\lambda}({\langle}e,\xi{\rangle})\,\mathrm{d}\sigma(\xi).$

Writing $f=f_{e}+f_{o}$, where $f_{e}$ is the even part and $f_{o}$ is the odd
part, we need to prove that $\hat{f}_{n}\geq 0$ when $f$ is either even or
odd. Since $C_{n}^{\lambda}$ is even if $n$ is even and odd if $n$ is odd,
$\hat{f}_{n}^{\lambda}=0$ if $fC_{n}^{\lambda}$ is odd. Hence, we only need to
show that $\hat{f}_{n}^{\lambda}\geq 0$ when $fC_{n}^{\lambda}$ is even. Now,
if $f$ is PDF on $\SS_{d+1,+}^{d}$, then so is $fC_{n}^{\lambda}$ by Schur’s
product theorem. Hence, applying (3.1) on $fC_{n}^{\lambda}$, we conclude that
$\hat{f}_{n}^{\lambda}\geq 0$. This completes the proof. ∎

We do not know if there is any element in $\Phi(\SS_{k,+}^{d})$ that is not of
the form (2.2).

Since an SPDF is automatically PDF, we now consider SPFs given by the Fourier-
Gegenbauer series (2.1) with $\hat{f}_{n}^{\lambda}\geq 0$, which can be
written as in (2.3) with the sum over the index set
${\mathcal{F}}=\\{n\in{\mathbb{N}}_{0}:\hat{f}_{n}>0\\}$ defined in (2.4).

######  Theorem 3.2.

Let $d\geq 2$ and ${\lambda}=\frac{d-1}{2}$. Let $k$ be an integer such that
$1\leq k\leq d+1$. A continuous function $f:[-1,1]\mapsto{\mathbb{R}}$ in
$\Phi(\SS^{d})$, or $\Phi(\SS_{d+1,+}^{d})$ if $k=d+1$, is strictly positive
definite if and only if ${\mathcal{F}}$ is an infinite set.

###### Proof.

Assume first that ${\mathcal{F}}$ is a finite set. Let $M$ be the largest
integer for which $\hat{f}_{n}\neq 0$. Since $C_{n}^{\lambda}$ is a polynomial
of degree $n$ and it has the same parity as $n$, we can write

$f(t)=\sum_{0\leq n\leq
M/2}\hat{f}_{M-2n}C_{M-2n}^{\lambda}(t)=\sum_{m=0}^{M/2}b_{m}t^{M-2m}$

for some set $(b_{m})$ of coefficients with $b_{0}\neq 0$. Let
$\Xi_{N}=\\{\xi_{1},\ldots,\xi_{N}\\}\subset\SS_{k,+}^{d}$. For
$\ell\in{\mathbb{N}}$, let $G_{\ell}$ be the matrix
$G_{\ell}=[{\langle}\xi_{i},\xi_{j}{\rangle}^{\ell}]_{i,j=1}^{N}$. For
$\ell=1$, $G_{\ell}$ is a Gram matrix, which is nonnegative definite and
$\mathrm{rank}\,G_{1}\leq d+1$, since $\xi_{j}\in\SS^{d}$. By Schur’s product
theorem, the matrices $G_{\ell}$ are also nonnegative definite and their ranks
are bounded by
$\mathrm{rank}\,G_{\ell}\leq(\mathrm{rank}\,G_{1})^{\ell}\leq(d+1)^{\ell}$.
Consequently, since
$\mathrm{rank}\,(A+B)\leq\mathrm{rank}\,A+\mathrm{rank}\,B$ for general
matrices $A$ and $B$, we conclude that

$\mathrm{rank}\,f[\Xi_{N}]\leq\sum_{m=0}^{M/2}\mathrm{rank}\,G_{M-2m}\leq\sum_{m=0}^{M/2}(d+1)^{M-2m},$

which is independent of $N$. Thus, for $N>\sum_{m=0}^{M/2}(d+1)^{M-2m}$, the
matrix $f[\Xi_{N}]$ does not have full rank and, hence, $f$ is not strictly
postive definite.

In the other direction, we assume that $f$ is a positive definite function
with an infinite set ${\mathcal{F}}$. For $k$ satisfying $1\leq k\leq d+1$,
let $\Xi_{N}=\\{\xi_{1},\ldots,\xi_{N}\\}$ be a set of $N$ distinct points in
$\SS_{k,+}^{d}$. Since $\Xi_{N}$ is a finite set, there are infinitely many
$p\in\SS_{k,+}^{d}$ such that none of the points $\xi_{i}-\xi_{j}$ are on the
hyper-plane defined by the condition ${\langle}\xi,p{\rangle}=0$, which means
that ${\langle}p,\xi_{i}{\rangle}\neq{\langle}p,\xi_{j}{\rangle}$ if $i\neq
j$. In particular, we fix such a $p$ such that ${\langle}p,\xi_{j}{\rangle}$
form a set of $N$ distinct numbers in $(-1,1)$; moreover, there exists no pair
$x_{i}$ and $x_{j}$ such that
${\langle}p,x_{i}{\rangle}=-{\langle}p,x_{j}{\rangle}$ due to
$\Xi_{N}\in\SS_{k,+}^{N}$. Let $\sigma$ be a rotation such that $\sigma
p=e_{d+1}=(0,\ldots,0,1)\in{\mathbb{R}}^{d}$. Considering the set
$\sigma\Xi_{N}=\\{\sigma\xi_{1},\ldots,\sigma\xi_{N}\\}$ if necessary, we can
assume $p=(0,\ldots,0,1)$. Thus, every point $\xi_{i}$ has a representation of
the form

$\xi_{i}=(\xi_{i}^{\prime}\sin{\theta}_{i},\cos{\theta}_{i}),\qquad\xi^{\prime}\in{\mathbb{S}^{d-1}},\quad\cos{\theta}_{i}={\langle}p,\xi_{i}{\rangle}.$

By the addition formula (2.7) of the Gegenbauer polynomials, we conclude that

$C_{n}^{\lambda}({\langle}\xi_{i},\xi_{j}{\rangle})=\sum_{k=0}^{n}b_{k,n}^{\lambda}Q_{k,n}^{\lambda}({\theta}_{i})Q_{k,n}^{\lambda}({\theta}_{j})C_{k}^{{\lambda}-\frac{1}{2}}({\langle}\xi_{i}^{\prime},\xi_{j}^{\prime}{\rangle}),$

which implies, in particular, that for
$\xi=(\xi^{\prime}\sin{\theta},\cos{\theta})\in\SS^{d}$,

$\sum_{j=1}^{N}c_{j}f({\langle}x,x_{j}{\rangle})=\sum_{n=0}^{\infty}\hat{f}_{n}^{\lambda}\left[\sum_{j=1}^{N}c_{j}Q_{k,n}^{\lambda}({\theta}_{j})C_{k}^{{\lambda}-\frac{1}{2}}({\langle}\xi_{j}^{\prime},\xi^{\prime}{\rangle})\right]Q_{k,n}^{\lambda}({\theta}).$

Since $\SS_{k,+}^{d}\subset\SS^{d}$, it is known that the matrix $f[\Xi_{N}]$
is positive definite if and only if the set of functions
$\\{f({\langle}\xi,\xi_{i}{\rangle}):1\leq i\leq N\\}$ is linearly independent
(cf. [10, Theorem 14.3.5]). Consequently, it is sufficient to prove that if

(3.2)
$\sum_{j=1}^{N}c_{j}Q_{k,n}^{\lambda}({\theta}_{j})C_{k}^{{\lambda}-\frac{1}{2}}({\langle}\xi_{j}^{\prime},\xi^{\prime}{\rangle})=0,\quad
0\leq k\leq n,\quad n\in{\mathcal{F}},$

then $c_{1}=c_{2}=\cdots=c_{n}=0$. This is proved by induction on $N$. It is
sufficient to consider (3.2) for $k=n$ since
$Q_{n,n}^{\lambda}({\theta}_{j})=(\sin{\theta}_{j})^{n}\neq 0$. The claim is
obvious true when $N=1$. Assume the claim has been proved for $M<N$. We now
prove that it holds for $\Xi_{N}$. Let $j_{0}$ be chosen such that
$\sin{\theta}_{j_{0}}=\max_{1\leq j\leq N}\sin{\theta}_{j}.$ Then
$\sin{\theta}_{j_{0}}>\sin{\theta}_{j}$ for $j\neq j_{0}$. Setting $k=n$ and
${\theta}={\theta}_{j_{0}}$ in (3.2) and dividing the equation by the
coefficient of $c_{j_{0}}$, we obtain

$\sum_{j\neq
j_{0}}c_{j}\left(\frac{\sin{\theta}_{j}}{\sin{\theta}_{j_{0}}}\right)^{n}\frac{C_{k}^{{\lambda}-\frac{1}{2}}({\langle}\xi_{j}^{\prime},\xi^{\prime}{\rangle})}{C_{k}^{{\lambda}-\frac{1}{2}}(1)}+c_{j_{0}}=0.$

Since ${\mathcal{F}}$ is an infinite set, let $n$ go to infinity through a
subset of integers in ${\mathcal{F}}$, we conclude, using the fac that
$|C_{n}^{{\lambda}-\frac{1}{2}}(t)|\leq Cn^{{\lambda}-\frac{1}{2}}(1)$ for
$t\in[-1,1]$ and $\sin{\theta}_{j}/\sin{\theta}_{j_{0}}\leq 1$ for $j\neq
k_{0}$, that $c_{j_{0}}=0$.Consquently, we can remove one point from (3.2)
and, by the induction hypothesis, conclude that $c_{j}=0$ for all $j$. The
proof is completed.

as the proof on $\SS^{d}$ when there are no pairs $\xi_{i}$ and $\xi_{j}$ for
which ${\langle}p,\xi_{i}{\rangle}=-{\langle}p,\xi_{j}{\rangle}$; see the
proof of Case 1 in [10, Theorem 14.3.6]. ∎

The characterization for $\mathrm{S}\Phi(\SS_{k,+}^{d})$ is different from
that for $\mathrm{S}\Phi(\SS^{d})$ given in Theorem 2.2. The main difference
lies in the existence of “antipodal pairs” of points in $\Xi_{N}$, which are
two points $\xi_{i}$ and $\xi_{j}$ such that $\xi_{i}=-\xi_{j}$. Such pairs
could exist in $\SS^{d}$ but cannot exist in $\SS_{k,+}^{\mathrm{d}}$. To see
the impact of such pairs, let us cosider, for example, the case that $f$ is an
even function and $\Xi_{N}$ contains a pair of antipodal points $\xi_{i}$ and
$\xi_{j}$; then the rows and columns corresponding to $\xi_{i}$ and $\xi_{j}$
in the matrix $f[\Xi_{N}]$ are identical, so that the matrix is not positive
definite. Thus, if $f$ is an even PDF function with an infinite
${\mathcal{F}}$ set, then it is not a SPDF on $\SS^{d}$, but it is on
$\SS_{+,d+1}^{d}$.

The proof of the Theorem 3.2 is a modification of the proof of Theorem 2.2
when no antipodal pairs exist; compare, for example, the proof given for [10,
Theorem 14.3.6]. We include the complete proof so that the paper is self-
contained.

##  4\. Positive definite functions on several regular domains

We consider PDFs when $\Omega$ is one of several regular domains, discussed in
the consecutive subsections. They contain balls and surfaces and solids of the
hyperbolic set, which should be contrasted with much of the earlier work we
cited in the introduction, much of which was on spheres.

###  4.1. PDF on the unit ball

On the unit ball ${\mathbb{B}}^{d}=\\{x\in{\mathbb{R}}^{d}:\|x\|\leq 1\\}$,
the distance function is given by

${\mathsf{d}}_{\mathbb{B}}(x,y)=\arccos\left({\langle}x,y{\rangle}+\sqrt{1-\|x\|^{2}}\sqrt{1-\|y\|^{2}}\right),$

which is a projection of the geodesic distance of the hemisphere
$\SS_{d+1,+}^{d}$ on ${\mathbb{B}}^{d}$. Thus, we see that a function
$f:[-1,1]\mapsto{\mathbb{R}}$ is a PDF on ${\mathbb{B}}^{d}$ if

$f[\Xi_{N}]=\left[f\left({\langle}x_{i},x_{j}{\rangle}+\sqrt{1-\|x_{i}\|^{2}}\sqrt{1-\|x_{j}\|^{2}}\right)\right]_{i,j=1}^{N}$

is nonnegative definite for all subsets
$\Xi=\\{x_{1},\ldots,x_{N}\\}\subset{\mathbb{B}}^{d}$. (For the moment, we do
not require these points in the ball to be distinct yet, this will begin to
play a rôle when we shall speak about strict positive definiteness.)

It is well-known that there is a one-to-one correspondence between
${\mathbb{B}}^{d}$ and the upper hemisphere $\SS_{d+1,+}^{d}$ defined by

(4.1) ${\mathbb{B}}^{d}\ni
x\mapsto\left(x,\sqrt{1-\|x\|^{2}}\right)\in\SS_{d+1,+}^{d},$

and the map preserves the distance between the two domains; more precisely,

${\mathsf{d}}_{{\mathbb{B}}}(x,y)={\mathsf{d}}_{\SS}\left(\left(x,\sqrt{1-\|x\|^{2}}\right),\left(y,\sqrt{1-\|y\|^{2}}\right)\right).$

Hence, the PDF functions on the unit ball are characterized by applying
Theorem 3.1 with $\SS_{d+1,+}^{d}$. Instead of stating the full analog of
Theorem 3.1, we state a simple version for the record.

######  Theorem 4.1.

Let $d\geq 2$. All functions $f:[-1,1]\mapsto{\mathbb{R}}$ in
$\Phi({\mathbb{B}}^{d})$ are given by the expansion

$f(\cos{\theta})=\sum_{n=0}^{\infty}a_{n}C_{n}^{\frac{d-1}{2}}(\cos{\theta}),\qquad
a_{n}\geq 0,\quad\forall n\in{\mathbb{N}}_{0}.$

Moreover, such an $f$ is in $\mathrm{S}\Phi({\mathbb{B}}^{d})$ if and only if
there are infinitely many positive $a_{n}$.

The ball is the first, and the simplest, of our regular domains $\Omega$. It
is worthwhile to discuss some relevant aspects of the connection to the unit
sphere in this simple case which has been studied before.

Under the mapping (4.1), the surface measure $\mathrm{d}\sigma(x)$ on the unit
sphere corresponds to $W_{0}(x)\,\mathrm{d}x$ on the unit ball, where the
positive function $W_{0}$ is the Chebyshev weight function

$W_{0}(x)=\frac{1}{\sqrt{1-\|x\|^{2}}},\qquad x\in{\mathbb{B}}^{d},$

on the unit ball. Let ${\mathcal{V}}_{n}({\mathbb{B}}^{d})$ be the space of
orthogonal polynomials of degree $n$ with respect to $W_{0}$ on
${\mathbb{B}}^{d}$. Let ${\mathbf{P}}_{n}(W_{0};\cdot,\cdot)$ be the
reproducing kernel of ${\mathcal{V}}_{n}({\mathbb{B}}^{d})$, as defined in
Definition 2.3. This kernel satisfies a closed-form formula, again called an
addition formula, given by (cf. [11, Corollary 4.2.9])

(4.2)
$\displaystyle{\mathbf{P}}_{n}(W_{0};x,y)=\frac{1}{2}\left[{\mathbf{P}}_{n}^{+}(W_{0};x,y)+{\mathbf{P}}_{n}^{-}(W_{0};x,y)\right],$

where ${\mathbf{P}}_{n}^{\pm}(W_{0};\cdot,\cdot)$ is defined by

${\mathbf{P}}_{n}^{\pm}(W_{0};x,y)=Z_{n}^{\frac{d-1}{2}}\left({\langle}x,y{\rangle}\pm\sqrt{1-\|x\|^{2}}\sqrt{1-\|y\|^{2}}\right).$

In particular,
${\mathbf{P}}_{n}^{+}(W_{0};x,y)=Z_{n}^{\frac{d-1}{2}}(\cos{\mathsf{d}}_{\mathbb{B}}(x,y))$.
It is not difficult to verify that the reproducing property (2.8) holds if
${\mathbf{P}}_{n}(W_{0};\cdot,\cdot)$ is replaced by either
${\mathbf{P}}_{n}^{+}(W_{0};\cdot,\cdot)$ or
${\mathbf{P}}_{n}^{-}(W_{0};\cdot,\cdot)$.

However, in order to be a reproducing kernel for
${\mathcal{V}}_{n}({\mathbb{B}}^{d})$, the kernel needs also satisfy the item
(2) of Definition 2.3 which requires that it be an element of
${\mathcal{V}}_{n}({\mathbb{B}}^{d})$ as a function of $x$ for each $y$. While
${\mathbf{P}}_{n}^{\pm}(W_{0};\cdot,\cdot)$ does not satisfy the latter
requirement, their sum ${\mathbf{P}}_{n}(W_{0};\cdot,\cdot)$ in (4.2) does.

Moreover, setting

$\xi=\left(x,\sqrt{1-\|x\|^{2}}\right)\quad\hbox{and}\quad\eta^{\pm}=\left(y,\pm\sqrt{1-\|y\|^{2}}\right),$

then the sum is the average of
$Z_{n}^{\frac{d-1}{2}}(\cos{\mathsf{d}}_{\SS}(\xi,\eta^{+}))$ and
$Z_{n}^{\frac{d-1}{2}}(\cos{\mathsf{d}}_{\SS}(\xi,\eta^{-}))$, where
$\eta^{-}$ is the reflect of $\eta^{+}$ over the plane defined by the
condition $\xi_{d+1}=0$. This connection of PDF to the reproducing kernel will
appear for the other domains that we shall discuss below as well.

The ball ${\mathbb{B}}^{d}$ is the projection of the upper hemisphere
$\SS_{d+1,+}^{d}$ under the mapping (4.1). We can also work with the lower
hemisphere $\SS_{d+1,-}=\\{\xi\in\SS^{d}:\xi_{d+1}\leq 0\\}$ using the map

$\SS^{d}\ni x\mapsto\left(x,-\sqrt{1-\|x\|_{2}}\right)\in\SS_{d+1,-}^{d},$

which again preserves the distance between the two sets.

It may be tempting to map a set of points on ${\mathbb{B}}^{d}$ to $\SS^{d}$
by using both mappings at the same time. For example, if $\Xi_{2N}$ is a set
of $2N$ distinct points in ${\mathbb{B}}^{d}$, such that
$\Xi_{2N}=\Xi_{N}^{+}\cup\Xi_{N}^{-}$ and $\Xi_{N}^{+}=-\Xi_{N}^{-}$, then we
can map the points in $\Xi_{N}^{+}$ to the upper hemisphere and points in
$\Xi_{N}^{-}$ to the lower hemisphere.

However, although both maps to hemispheres preserve the distance, the total
collection of points does not. Indeed, if $x_{i}\mapsto
X_{i}=(x_{i},\sqrt{1-\|x_{i}\|^{2}})\in\SS_{d+1,+}^{d}$ and $x_{j}\mapsto
X_{j}=(x_{j},-\sqrt{1-\|x_{j}\|^{2}})\in\SS_{d+1,-}^{d}$ are different points,
then

$\cos{\mathsf{d}}_{\SS}(X_{i},X_{j})={\langle}x_{i},x_{j}{\rangle}-\sqrt{1-\|x_{i}\|^{2}}\sqrt{1-\|x_{j}\|^{2}}\neq\cos{\mathsf{d}}_{\mathbb{B}}(x_{i},x_{j}).$

###  4.2. PDF on the hyperbolic surface

Up to an affine change of variable, the hyperbolic surface that we shall
consider in this subsection can be normalized to satisfy, for a constant
$\varrho\geq 0$,

${}_{\varrho}{\mathbb{X}}_{0}^{d+1}:=\left\\{(x,t):\|x\|=\sqrt{t^{2}-\varrho^{2}},\quad
x\in{\mathbb{R}}^{d},\,\varrho\leq|t|\leq\sqrt{1+\varrho^{2}}\right\\},$

which degenerates to a double cone if $\varrho=0$. The surface is naturally
divided into upper and lower hyperbolic surfaces, which we denote by

${}_{\varrho}{\mathbb{X}}_{0}^{d+1}={}_{\varrho}{\mathbb{X}}_{0,+}^{d+1}\cup{}_{\varrho}{\mathbb{X}}_{0,-}^{d+1},\quad\hbox{where}\quad{}_{\varrho}{\mathbb{X}}_{0,\pm}^{d+1}=\left\\{(x,t)\in{}_{\varrho}{\mathbb{X}}_{0}^{d+1}:\mathrm{sign}(t)=\pm
1\right\\}.$

For $\varrho>0$, the upper and lower parts are disjoint, so we only consider
the distance for points in either the upper or the lower hyperbolic surfaces.
For $(x,t),(y,s)$ both in ${}_{\varrho}{\mathbb{X}}_{0,+}^{d+1}$ or both in
${}_{\varrho}{\mathbb{X}}_{0,-}^{d+1}$, their distance is defined by [22]

${\mathsf{d}}_{{\mathbb{X}}_{0}}^{\varrho}\big{(}(x,t),(y,s)\big{)}=\arccos\left({\langle}x,y{\rangle}+\sqrt{1+\rho^{2}-t^{2}}\sqrt{1+\varrho^{2}-s^{2}}\right).$

In particular, for $\Xi_{N}=\\{(x_{j},t_{j}):1\leq i\leq
N\\}\subset{}_{\varrho}{\mathbb{X}}_{0}^{d+1}$, the matrix $f[\Xi_{N}]$ is
given by

$f[\Xi_{N}]=\left[f\left({\langle}x_{i},x_{j}{\rangle}+\sqrt{1+\varrho^{2}-t_{i}^{2}}\sqrt{1+\varrho^{2}-t_{j}^{2}}\right)\right]_{i,j=1}^{N}.$

Like the case of the unit ball, there is a one-to-one correspondence between
${}_{\varrho}{\mathbb{X}}_{0,+}^{d+1}$ and the upper hemisphere
$\SS_{d+1,+}^{d}$, defined by

(4.3)
${}_{\varrho}{\mathbb{X}}_{0,+}^{d+1}\ni(x,t)\mapsto\left(x,\sqrt{1+\varrho^{2}-t^{2}}\right)\in\SS_{d+1,+}^{d},$

that preserves the distance on the domains; that is,

${\mathsf{d}}_{{\mathbb{X}}_{0}}^{\varrho}\big{(}(x,t),(y,s)\big{)}={\mathsf{d}}_{\SS}\left(\left(x,\sqrt{1+\varrho^{2}-t^{2}}\right),\left(y,\sqrt{1+\varrho^{2}-s^{2}}\right)\right).$

Moreover, the same mapping also defines a correspondence between
${\mathbb{X}}_{0,-}^{d+1}$ and $\SS_{d+1,+}^{d}$.

As a result, we obtain a characterization of the positive definite functions
on the upper, or lower, hyperbolic surface from Theorem 3.1. As in the case of
the unit ball, we state a simplified version for the record.

######  Theorem 4.2.

Let $d\geq 2$. All functions $f:[-1,1]\mapsto{\mathbb{R}}$ in
$\Phi({}_{\varrho}{\mathbb{X}}_{0,\pm}^{d+1})$ are given by the expansion

$f(\cos{\theta})=\sum_{n=0}^{\infty}a_{n}C_{n}^{\frac{d-1}{2}}(\cos{\theta}),\qquad
a_{n}\geq 0,\quad\forall n\in{\mathbb{N}}_{0}.$

Moreover, such an $f$ is in $\mathrm{S}\Phi({\mathbb{B}}^{d})$ if and only if
there are infinitely many positive $a_{n}$.

Under the mapping (4.3), the surface measure $\mathrm{d}\sigma(t)$ on
$\SS^{d}$ is mapped to

$w_{0}(t)\,\mathrm{d}\sigma_{{\mathbb{X}}_{0}}(x,t),$

where $w_{0}$ is the Chebyshev weight $w_{0}(t)=1/\sqrt{1-t^{2}}$ and
$\mathrm{d}\sigma_{{\mathbb{X}}_{0}}$ denotes the Lebesgue measure on
${}_{\varrho}{\mathbb{X}}_{0}^{d+1}$. The orthogonal polynomials in
$L^{2}({}_{\varrho}{\mathbb{X}}_{0}^{d+1},w_{0})$ are divided into two
families, one contains those that are even in the $t$ variable and the other
contains those that are odd in the $t$ variable [20]. Let
${\mathcal{V}}_{n}^{E}({}_{\varrho}{\mathbb{X}}_{0}^{d+1})$ be the space of
orthogonal polynomials of degree $n$ that are even in the $t$ variable. An
explicit orthogonal basis of the space is given in [20]. Let
${\mathsf{P}}_{n}^{E}(w_{0};\cdot,\cdot)$ be the reproducing kernel of
${\mathcal{V}}_{n}^{E}({}_{\varrho}{\mathbb{X}}_{0}^{d+1})$ in
$L^{2}({}_{\varrho}{\mathbb{X}}_{0}^{d+1},w_{0})$. The kernel satisfies an
addition formula given by [20, (3.5)]

(4.4)
$\displaystyle{\mathsf{P}}_{n}^{E}\big{(}w_{0};(x,t),(y,s)\big{)}=\frac{1}{2}\left[{\mathsf{P}}_{n}^{+}\big{(}w_{0};(x,t),(y,s)\big{)}+{\mathsf{P}}_{n}^{-}\big{(}w_{0};(x,t),(y,s)\big{)}\right],$

where ${\mathsf{P}}_{n}^{\pm}\big{(}w_{0};\cdot,\cdot\big{)}$ is defined by

${\mathsf{P}}_{n}^{\pm}\big{(}w_{0};(x,t),(y,s)\big{)}=Z_{n}^{\frac{d-1}{2}}\left({\langle}x,y{\rangle}\mathrm{\,sign}(ts)\pm\sqrt{1-t^{2}}\sqrt{1-s^{2}}\right).$

If $(x,t)$ and $(y,s)$ are both in ${}_{\varrho}{\mathbb{X}}_{0,+}^{d+1}$ or
both in ${}_{\varrho}{\mathbb{X}}_{0,-}^{d+1}$, then
${\mathsf{P}}_{n}^{+}\big{(}w_{0};(x,t),(y,s)\big{)}=Z_{n}^{\frac{d-1}{2}}\big{(}{\mathsf{d}}_{{\mathbb{X}}_{0}}^{\varrho}((x,t),(y,s))\big{)}$.
Like the case in the unit ball, ${\mathsf{P}}_{n}^{\pm}(w_{0};\cdot,\cdot)$
satisfies the reproducing property, but it is not a polynomial, whereas the
sum in (4.4) is.

###  4.3. PDF on the solid hyperboloid

For $\varrho\geq 0$, the solid hyperboloid is bounded by the hyperbolic
surface ${}_{\varrho}{\mathbb{X}}_{0}^{d+1}$ and hyperplanes defined by the
condition $t=\pm\sqrt{1+\varrho^{2}}$,

${}_{\varrho}{\mathbb{X}}^{d+1}:=\left\\{(x,t):\|x\|\leq\sqrt{t^{2}-\varrho^{2}},\quad
x\in{\mathbb{R}}^{d},\,\varrho\leq|t|\leq\sqrt{1+\varrho^{2}}\right\\},$

which degenerates to the double cone if $\varrho=0$. The hyperboloid is
naturally divided into upper and lower part, which we denote by

${}_{\varrho}{\mathbb{X}}^{d+1}={}_{\varrho}{\mathbb{X}}_{+}^{d+1}\cup{}_{\varrho}{\mathbb{X}}_{-}^{d+1},\quad\hbox{where}\quad{}_{\varrho}{\mathbb{X}}_{\pm}^{d+1}=\left\\{(x,t)\in{}_{\varrho}{\mathbb{X}}^{d+1}:\mathrm{sign}(t)=\pm
1\right\\}.$

Like the case of hyperbolic surface, we only consider the distance for points
in either the upper or the lower hyperboliod. For $(x,t),(y,s)$ both in
${}_{\varrho}{\mathbb{X}}_{+}^{d+1}$ or both in
${}_{\varrho}{\mathbb{X}}_{-}^{d+1}$, their distance is defined by [22,
Definition 4.1 and (4.6)]

$\displaystyle{\mathsf{d}}_{{\mathbb{X}}}^{\varrho}\big{(}(x,t),(y,s)\big{)}=\arccos$
$\displaystyle\left({\langle}x,y{\rangle}+\sqrt{t^{2}-\varrho^{2}-\|x\|^{2}}\sqrt{s^{2}-\varrho^{2}-\|y\|^{2}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\sqrt{1+\rho^{2}-t^{2}}\sqrt{1+\varrho^{2}-s^{2}}\right),$

which is a mixture of the distances on the unit ball and on the hyperbolic
surface. There is a one-to-one correspondence between
${}_{\varrho}{\mathbb{X}}_{+}^{d+1}$ and
$\SS_{d,+}^{d}=\\{\xi\in\SS^{d}:\xi_{d}\geq 0,\xi_{d+1}\geq 0\\}$, defined by

(4.5)
${}_{\varrho}{\mathbb{X}}_{+}^{d+1}\ni(x,t)\mapsto\left(x,\sqrt{t^{2}-\varrho^{2}-\|x\|^{2}},\sqrt{1+\varrho^{2}-t^{2}}\right)=:\xi(x,t)\in\SS_{d,+}^{d}.$

This preserves the distance between the domains; that is,

${\mathsf{d}}_{{\mathbb{X}}}^{\varrho}\big{(}(x,t),(y,s)\big{)}={\mathsf{d}}_{\SS}\big{(}\xi(x,t),\xi(y,s)\big{)}.$

Likewise, we can define an analog correspondence between
${}_{\varrho}{\mathbb{X}}_{-}^{d+1}$ and $\SS_{d,+}^{d}$. Consequently, we
obtain a characterization of the positive definite functions on the upper, or
lower, hyperboloid from Theorem 3.1 for $\SS_{d,+}^{d}$. Again, we record a
simplified version of the characterization.

######  Theorem 4.3.

Let $d\geq 1$. All functions $f:[-1,1]\mapsto{\mathbb{R}}$ given by the
expansion

$f(\cos{\theta})=\sum_{n=0}^{\infty}a_{n}C_{n}^{\frac{d-1}{2}}(\cos{\theta}),\qquad
a_{n}\geq 0,\quad\forall n\in{\mathbb{N}}_{0}.$

are in $\Phi({}_{\varrho}{\mathbb{X}}_{\pm}^{d+1})$. And such an $f$ is in
$\mathrm{S}\Phi({}_{\varrho}{\mathbb{X}}_{\pm}^{d+1})$ if and only if there
are infinitely many positive $a_{n}$.

Under the mapping (4.5), the measure on ${}_{\varrho}{\mathbb{X}}_{+}^{d+1}$
that corresponds to the surface measure $\mathrm{d}\sigma$ on $\SS^{d}$ is
$W_{0,0}(x,t)\,\mathrm{d}x\,\mathrm{d}t$, where $W_{0,0}$ is given by

$W_{0,0}(x,t)=\frac{|t|}{\sqrt{1-t^{2}}\sqrt{1-\|x\|^{2}}}.$

As in the case of hyperbolic surface, the orthogonal polynomials in
$L^{2}({}_{\varrho}{\mathbb{X}}^{d+1},W_{0,0})$ are divided into two families,
depending on their parity in the $t$ variable. Let
${\mathcal{V}}_{n}^{E}({}_{\varrho}{\mathbb{X}}^{d+1})$ be the space of
orthogonal polynomials of degree $n$ that are even in the $t$ variable. Then
the reproducing kernel ${\mathbf{P}}_{n}^{E}(W_{0,0};\cdot,\cdot)$ of
${\mathcal{V}}_{n}^{E}({}_{\varrho}{\mathbb{X}}^{d+1})$ satisfies an addition
formula given by [20, Corollary 5.6 and (5.16)]

(4.6)
$\displaystyle{\mathbf{P}}_{n}^{E}\big{(}W_{0,0};(x,t),(y,s)\big{)}=\frac{1}{4}$
$\displaystyle\left[{\mathbf{P}}_{n}^{+,+}\big{(}(x,t),(y,s)\big{)}+{\mathbf{P}}_{n}^{+,-}\big{(}(x,t),(y,s)\big{)}\right.$
$\displaystyle\left.+{\mathbf{P}}_{n}^{-,+}\big{(}(x,t),(y,s)\big{)}+{\mathbf{P}}_{n}^{-,-}\big{(}(x,t),(y,s)\big{)}\right],$

where ${\mathbf{P}}_{n}^{\pm,\pm}(\cdot,\cdot)$ is defined by

$\displaystyle{\mathbf{P}}_{n}^{\pm,\pm}\big{(}(x,t),(y,s)\big{)}$
$\displaystyle=Z_{n}^{\frac{d-1}{2}}\left(\left({\langle}x,y{\rangle}\pm\sqrt{t^{2}-\varrho^{2}-\|x\|^{2}}\sqrt{s^{2}-\varrho^{2}-\|y\|^{2}}\right)\mathrm{sign}(ts)\right.$
$\displaystyle\hskip
137.31255pt\left.\pm\sqrt{1+\varrho^{2}-t^{2}}\sqrt{1+\varrho^{2}-s^{2}}\right).$

As in the previous cases, the kernel ${\mathbf{P}}_{n}^{+,+}(\cdot,\cdot)$ is
the same as
$Z_{n}^{\frac{d-1}{2}}(\cos{\mathsf{d}}_{{\mathbb{X}}}^{\varrho}((x,t),(y,s)))$
so that it is a positive definite kernel on the upper, or lower, hyperboloid
${}_{\varrho}{\mathbb{X}}_{\pm}^{d+1}$, and it satisfies the reproducing
property. The reproducing kernel given in (4.6) comes from the average that
makes it a kernel of polynomials. Furthermore, the average is over
$Z_{n}^{\frac{d-1}{2}}\big{(}\cos{\mathsf{d}}_{\SS}^{\varrho}(\xi(x,t),\xi^{\pm,\pm}(y,s))\big{)}$,
where $\xi^{\pm,\pm}=(\xi_{1},\ldots,\xi_{d-1},\pm\xi_{d},\pm\xi_{d+1})$ are
reflections of $\xi$ with respect to the hyperplanes $\xi_{d}=0$ and
$\xi_{x+1}=0$.

###  4.4. PDF on the conic surface

Up to an affine transformation, we consider the conic surface defined by

${\mathbb{V}}_{0}^{d+1}:=\left\\{(x,t):\|x\|=t,\;x\in{\mathbb{B}}^{d},\;t\in[0,1]\right\\}.$

The distance function in this domain is defined by [21, (4.1)]

${\mathsf{d}}_{{\mathbb{V}}_{0}}\big{(}(x,t),(y,s)\big{)}=\arccos\left(\sqrt{\frac{ts+{\langle}x,y{\rangle}}{2}}+\sqrt{1-t}\sqrt{1-s}\right),$

which is different from that on the double cone
${}_{0}{\mathbb{X}}_{0}^{d+1}$. We do not know if there is a distance
preserving map from ${\mathbb{V}}_{0}^{d+1}$ to a subset of $\SS^{d}$ for
$d>2$. For $d=2$, however, we have such a map between ${\mathbb{V}}_{0}^{3}$
and $\SS_{2,+}^{2}$. The latter is given by

$\SS_{2,+}^{2}=\\{(\xi_{1},\xi_{2},\xi_{3})\in\SS^{2}:\xi_{2}\geq
0,\xi_{3}\geq 0\\},$

which is the half of the upper hemisphere in ${\mathbb{R}}^{3}$.

######  Proposition 4.4.

The mapping $\xi:{\mathbb{V}}_{0}^{3}\mapsto\SS_{2,+}^{2}$ defined by

(4.7)
$\xi:(x_{1},x_{2},t)\in{\mathbb{V}}_{0}^{3}\mapsto\left(\sqrt{\frac{t-x_{1}}{2}}\,\mathrm{sign}(x_{2}),\sqrt{\frac{t+x_{1}}{2}},\sqrt{1-t}\right)=:\xi(x,t)\in\SS^{2}_{2,+}$

is a one-to-one correspondence and it preserves the distances between the
domains; that is,

${\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))=\arccos\left({\langle}\xi(x,t),\xi(y,s){\rangle}\right)={\mathsf{d}}_{\SS}\left(\xi(x,t),\xi(y,s)\right).$

###### Proof.

It is easy to verify that $\xi_{x,t}\in\SS_{2,+}^{2}$. The inverse of the map
is given by

$\SS_{2,+}^{2}\ni\xi\mapsto\left(\xi_{2}^{2}-\xi_{1}^{2},2\xi_{1}\xi_{2},1-\xi_{3}^{2}\right)=(x_{1},x_{2},t)\in{\mathbb{V}}_{0}^{3}.$

To see that the mapping preserves the distance, we use the following identity

$\frac{1}{4}\left(\sqrt{(t+x_{1})(s+y_{1})}+\mathrm{sign}(x_{2}y_{2})\sqrt{(t-x_{1})(s-y_{1})}\right)^{2}=\frac{1}{2}(ts+x_{1}y_{1}+x_{2}y_{2}),$

which was used to prove that ${\mathsf{d}}_{{\mathbb{V}}_{0}}$ is indeed a
distance on ${\mathbb{V}}_{0}^{3}$ in [21, Proposition 4.2], where, however,
the factor $\frac{1}{4}$ was missing. ∎

A point on ${\mathbb{V}}_{0}^{3}$ has three components, which we write as
$(x,y,t)$ for $(x,y)\in{\mathbb{B}}^{2}$ and $t\in[0,1]$. For a set of points
$\Xi_{N}=\\{(x_{i},y_{i},t_{i}),1\leq i\leq N\\}$ in ${\mathbb{V}}_{0}^{3}$,
the matrix $f[\Xi_{N}]$ is given by

$f[\Xi_{N}]=\left[\sqrt{\frac{t_{i}t_{j}+x_{i}x_{j}+y_{i}y_{j}}{2}}+\sqrt{1-t_{i}}\sqrt{1-t_{j}}\right]_{i,j=1}^{N}.$

Note that the elements of the matrix are all nonnegative. Since
$\cos\mathrm{d}_{{\mathbb{V}}_{0}}((x,t),(y,s))\geq 0$, we only need the
function $f$ on $[0,1]$. Instead, we shall assme that $f$ is an even function
on $[-1,1]$. Then $f$ can be expanded as a series of
$C_{n}^{\frac{1}{2}}(t)=P_{n}(t)$, where the $P_{n}$ are the Legendre
polynomials. Using the distance preserving correspondence of
${\mathbb{V}}_{0}^{3}$ and $\SS_{2,+}^{2}$, we can derive the characterization
of PDFs on ${\mathbb{V}}_{0}^{3}$ from Theorem 3.1 with $\SS^{2}_{2,+}$.

######  Theorem 4.5.

All functions $f:[-1,1]\mapsto{\mathbb{R}}$ that are even on $[-1,1]$ and
given by the expansion

$f(\cos{\theta})=\sum_{n=0}^{\infty}a_{2n}P_{2n}(\cos{\theta}),\qquad
a_{2n}\geq 0,\quad\forall n\in{\mathbb{N}}_{0}$

are in $\Phi({\mathbb{V}}_{0}^{3})$. Moreover, such an $f$ is in
$\mathrm{S}\Phi({\mathbb{V}}^{3})$ if and only if there are infinitely many
positive $a_{2n}$.

On the ${\mathbb{V}}_{0}^{d+1}$, the Chebyshev weight function is $w_{-1}$,
defined by

$w_{-1}(t)=t^{-1}(1-t)^{-\frac{1}{2}},\qquad 0\leq t\leq 1.$

The orthogonal polynomials with respect to
$w_{-1}(t)\,\mathrm{d}\sigma_{{\mathbb{V}}_{0}}(x,t)$ are studied in [19]. Let
${\mathcal{V}}_{n}({\mathbb{V}}_{0}^{3})$ be the space of orthogonal
polynomials of degree $n$ in $L^{2}({\mathbb{V}}_{0}^{3},w_{-1})$. The
reproducing kernel ${\mathsf{P}}_{n}(w_{-1};\cdot,\cdot)$ of the space
${\mathcal{V}}_{n}({\mathbb{V}}_{0}^{3})$ satisfies an addition formula given
by [19, (8.6)],

(4.8)
$\displaystyle{\mathsf{P}}_{n}\big{(}w_{-1};(x,t),(y,s)\big{)}=\frac{1}{4}$
$\displaystyle\left[{\mathsf{P}}_{n}^{+,+}\big{(}(x,t),(y,s)\big{)}+{\mathsf{P}}_{n}^{+,-}\big{(}(x,t),(y,s)\big{)}\right.$
$\displaystyle\left.{}+{\mathsf{P}}_{n}^{-,+}\big{(}(x,t),(y,s)\big{)}+{\mathsf{P}}_{n}^{-,-}\big{(}(x,t),(y,s)\big{)}\right],$

where ${\mathsf{P}}_{n}^{\pm,\pm}\big{(}\cdot,\cdot\big{)}$ is defined by,
with $x=(x_{1},x_{2})$ and $y=(y_{1},y_{2})$,

$\displaystyle{\mathsf{P}}_{n}^{\pm,\pm}\big{(}(x,t),(y,s)\big{)}=Z_{2n}^{\frac{1}{2}}\left(\sqrt{\frac{ts\pm(x_{1}y_{1}+x_{2}y_{2})}{2}}\pm\sqrt{1-t}\sqrt{1-s}\right).$

Recall that $Z_{n}^{\frac{1}{2}}(t)=(2n+1)C_{n}^{\frac{1}{2}}=(2n+1)P_{n}(t)$.
Thus, once again, we see that ${\mathsf{P}}_{n}^{+,+}(\cdot,\cdot)$ is a
positive definite kernel on ${\mathbb{V}}_{0}^{3}$ and the sum in (4.8) is an
average that makes it a kernel of polynomials. Moreover, for each fixed
$(x,t)$, the average is taken over
$Z_{n}^{\frac{1}{2}}\big{(}\cos{\mathsf{d}}_{\SS}(\xi(x,t)),\xi^{\pm}(y,s)\big{)}$,
where $\xi^{\pm}=(\xi_{1},\pm\xi_{2},\pm\xi_{3})$.

Finally, let us mention that we do not know how to characterize the PDFs on
the solid cone ${\mathbb{V}}^{3}$ bounded by ${\mathbb{V}}_{0}^{3}$ and the
hyperplane $t=1$. Making use of the characterization of
$\Phi({\mathbb{V}}_{0}^{3})$, we can obtain a characterization of
$\Phi({\mathbb{V}}_{0}^{2})$. However, ${\mathbb{V}}_{0}^{2}$ is a triangle,
which becomes the triangle ${\mathbb{T}}^{2}$ in the next subsection by an
affine transformation.

###  4.5. PDF on the simplex

The simplex ${\mathbb{T}}^{d}$ of ${\mathbb{R}}^{d}$ is defined by

${\mathbb{T}}^{d}=\left\\{x\in{\mathbb{R}}^{d}:x_{1}\geq 0,\ldots,x_{d}\geq
0,\,|x|\leq 1\right\\},$

where $|x|=x_{1}+\cdots+x_{d}$. The distance function on this domain is
defined by

${\mathsf{d}}_{\mathbb{T}}(x,y)=\arccos\left(\left\langle\sqrt{x},\sqrt{y}\right\rangle+\sqrt{1-|x|}\sqrt{1-|y|}\right),$

where $\sqrt{x}:=(\sqrt{x_{1}},\ldots,\sqrt{x_{d}})$. For a set of points
$\Xi_{N}=\\{x_{i}:1\leq i\leq N\\}$ in ${\mathbb{T}}^{d}$, the matrix
$f[\Xi_{N}]$ is then given by

$f[\Xi_{N}]=\left[\left\langle\sqrt{x_{i}},\sqrt{x_{j}}\right\rangle+\sqrt{1-|x_{i}|}\sqrt{1-|x_{j}|}\right]_{i,j=1}^{N},$

which is a matrix that has all of its elements nonnegative. It is known that
there is a one-to-one correspondence between ${\mathbb{T}}^{d}$ and the
positive quadrant of the unit sphere $\SS_{1,+}^{d}$, which is defined by

${\mathbb{T}}^{d}\ni
x\mapsto\left(\sqrt{x_{1}},\ldots,\sqrt{x_{d}},\sqrt{1-|x|}\right)=T(x)\in\SS_{1,+}^{d}$

and it preserves the distance between the two domains:

${\mathsf{d}}_{\mathbb{T}}(x,y)=\arccos{\langle}T(x),T(y){\rangle}={\mathsf{d}}_{\SS}\left(T(x),T(y)\right).$

Hence, the characterization of PDF on the simplex follows from Theorem 3.1. As
is the case on the conic surface, $\cos{\mathsf{d}}_{\mathbb{T}}(x,y)$ is
nonnegative so we only need $f$ to be supported on $[0,1]$, which we extended
evenly to $[-1,1]$. As in all previous cases in the section, we state a
simplified version for the record.

######  Theorem 4.6.

All functions $f:[-1,1]\mapsto{\mathbb{R}}$ that are even on $[-1,1]$ and
given by the expansion

$f(\cos{\theta})=\sum_{n=0}^{\infty}a_{2n}C_{2n}^{\frac{d-1}{2}}(\cos{\theta}),\qquad
a_{2n}\geq 0,\quad\forall n\in{\mathbb{N}}_{0}$

are in $\Phi({\mathbb{T}}^{d})$. Moreover, such an $f$ is in
$\mathrm{S}\Phi({\mathbb{T}}^{d})$ if and only if there are infinitely many
positive $a_{2n}$.

The weight function on the simplex ${\mathbb{T}}^{d}$ that corresponds to the
surface measure is the Chebyshev weight function

$W_{0}(x)=\frac{1}{\sqrt{x_{1}\cdots x_{d}(1-|x|)}}.$

Let ${\mathcal{V}}_{n}({\mathbb{T}}^{d})$ be the space of orthogonal
polynomials in $L^{2}({\mathbb{T}}^{d},W_{0})$. Then the reproducing kernel
${\mathbf{P}}_{n}(W_{0};\cdot,\cdot)$ satisfies an addition formula given by
[11, (4.4.5)]

${\mathbf{P}}_{n}(W_{0};x,y)=\frac{1}{2^{d}}\sum_{{\varepsilon}\in\\{1,-1\\}^{d}}Z_{2n}^{\frac{d-1}{2}}\left(\left\langle\sqrt{x},{\varepsilon}\sqrt{y}\right\rangle+\sqrt{1-|x|}\sqrt{1-|y|^{2}}\right).$

As in all previous cases, the kernel is derived from the positive definite
kernel

$Z_{2n}^{\frac{d-1}{2}}(\cos{\mathsf{d}}_{\mathbb{T}}(x,y))$

by taking an average that ensures it is a kernel of polynomials. Moreover, the
average is taken over
$Z_{2n}^{\frac{d-1}{2}}(\cos{\mathsf{d}}_{\mathbb{T}}(x,{\varepsilon}y))$,
where ${\varepsilon}y$ is the reflection of $y$ over a coordinate plane.

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