# Sampling in $\Lambda$-shift-invariant subspaces of Hilbert-Schmidt operators
on $L^{2}(\mathbb{R}^{d})$

Antonio G. García<EMAIL_ADDRESS>
###### Abstract

The translation of an operator is defined by using conjugation with time-
frequency shifts. Thus, one can define $\Lambda$-shift-invariant subspaces of
Hilbert-Schmidt operators, finitely generated, with respect to a lattice
$\Lambda$ in $\mathbb{R}^{2d}$. These spaces can be seen as a generalization
of classical shift-invariant subspaces of square integrable functions.
Obtaining sampling results for these subspaces appears as a natural question
that can be motivated by the problem of channel estimation in wireless
communications. These sampling results are obtained in the light of the frame
theory in a separable Hilbert space.

  * Departamento de Matemáticas, Universidad Carlos III de Madrid, Spain.

Keywords: Hilbert-Schmidt operators; Weyl transform; Kohn-Nirenberg transform;
Translation of operators; $\Lambda$-shift-invariant subspaces; Sampling
Hilbert-Schmidt operators.

AMS: 42C15; 43A32; 47B10; 94A20.

##  1 Introduction

In this paper we obtain sampling results in shift-invariant-like subspaces of
the class $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ of Hilbert-Schmidt
operators on $L^{2}(\mathbb{R}^{d})$. To be more precise, these subspaces are
obtained by translation in a lattice $\Lambda\subset\mathbb{R}^{2d}$ of a
fixed set of Hilbert-Schmidt operators $S_{1},S_{2},\dots,S_{N}$. The
translation of an operator $S$ by $z\in\mathbb{R}^{2d}$ is defined by using
conjugation with the time-frequency shift $\pi(z)$, where $z=(x,\omega)$
belongs to the phase space $\mathbb{R}^{d}\times\widehat{\mathbb{R}}^{d}$
(which in the sequel will be identified with $\mathbb{R}^{2d}$) by

$\alpha_{z}(S):=\pi(z)S\,\pi(z)^{*}\,,\quad z\in\mathbb{R}^{2d}\,.$

Recall that the time-frequency shift acts on $f\in L^{2}(\mathbb{R}^{d})$ as
$\pi(z)f(t)={\rm e}^{2\pi i\omega\cdot t}f(t-x)$. The set of translations
$\\{\alpha_{z}\\}_{z\in\mathbb{R}^{2d}}$ is a unitary representation of the
group $\mathbb{R}^{2d}$ on the Hilbert space
$\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$.

If we take a full rank lattice $\Lambda$ in $\mathbb{R}^{2d}$, i.e.,
$\Lambda=A\mathbb{Z}^{d}$ where $A$ is a $2d\times 2d$ real invertible matrix,
such that the sequence
$\\{\alpha_{\lambda}(S_{n})\\}_{\lambda\in\Lambda;\,n=1,2,\dots,N}$ is a Riesz
sequence in $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ we consider the subspace
of $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ given by

$V_{\bf
S}^{2}=\Big{\\{}\sum_{n=1}^{N}\sum_{\lambda\in\Lambda}c_{n}(\lambda)\,\alpha_{\lambda}(S_{n})\,\,:\,\,\\{c_{n}(\lambda)\\}_{\lambda\in\Lambda}\in\ell^{2}(\Lambda)\,,\,n=1,2,\dots,N\Big{\\}}\,.$

From now on, the subspaces $V_{\bf S}^{2}$ obtained in this way will be called
$\Lambda$-shift-invariant subspaces in
$\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$. These spaces are a generalization of
the classical shift-invariant subspaces in $L^{2}(\mathbb{R}^{d})$:

$V^{2}_{\Phi}:=\Big{\\{}\sum_{n=1}^{N}\sum_{\alpha\in\mathbb{Z}^{d}}c_{n}(\alpha)\,\varphi_{n}(t-\alpha)\,\,:\,\,\\{c_{n}(\alpha)\\}_{\alpha\in\mathbb{Z}^{d}}\in\ell^{2}(\mathbb{Z}^{d})\,,\,n=1,2,\dots,N\Big{\\}}\,,$

where $\Phi=\\{\varphi_{1},\varphi_{2},\dots,\varphi_{{}_{N}}\\}$ denotes a
set of generators of $V^{2}_{\Phi}$. Sampling in the shift-invariant subspace
$V^{2}_{\Phi}$ usually involves, for each $f\in V^{2}_{\Phi}$, pointwise
samples $\\{f(\alpha+\beta_{m})\\}_{\alpha\in\mathbb{Z}^{d}}$ and/or average
samples $\\{\langle
f,\psi_{m}(\cdot-\alpha)\rangle\\}_{\alpha\in\mathbb{Z}^{d}}$, where
$\psi_{m}$ is an average function in $L^{2}(\mathbb{R}^{d})$, which not
necessarily belong to $V_{\varphi}^{2}$. Any stable sampling in $V^{2}_{\Phi}$
will involve, necessarily, $M\geq N$ sequences of samples (see, for instance,
[1, 13] and references therein).

A challenge problem here is to choose an appropriate set of samples that
should be used for operators in $V_{\bf S}^{2}$. Inspired in Ref. [18] and
motivated by the problem of channel estimation in wireless communications, in
this paper we propose for any $T\in V_{\bf S}^{2}$ its diagonal channel
samples at the lattice $\Lambda\subset\mathbb{R}^{2d}$ defined by

$s_{{}_{T,m}}(\lambda):=\big{\langle}\alpha_{{}_{-\lambda}}(T)g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\,,\quad\lambda\in\Lambda\,,\,\,\,m=1,2,\dots,M\,,$
(1)

where $g_{m},\widetilde{g}_{m}$, $m=1,2,\dots,M$, are $2M$ fixed functions in
$L^{2}(\mathbb{R}^{d})$ (we will see that necessarily $M\geq N$). The name
diagonal channel samples coined for these samples will become clear later on
where a little explanation will be done for both, the choice of Hilbert-
Schmidt operators (in $V_{\bf S}^{2}$) to be sampled, and the choice of the
above samples for any $T\in V_{\bf S}^{2}$. As we will see in Section 3.3 the
samples defined in (1) are nothing but the lower symbol of the operator $T$
with respect $g_{m},\widetilde{g}_{m}\in L^{2}(\mathbb{R}^{d})$ and lattice
$\Lambda$, i.e.,
$\big{\langle}T\pi(\lambda)g_{m},\pi(\lambda)\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}$,
$\lambda\in\Lambda$, or the samples of the Berezin transform
$\mathcal{B}^{g_{m},\widetilde{g}_{m}}\,T(z):=\big{\langle}T\pi(z)g_{m},\pi(z)\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}$,
$z\in\mathbb{R}^{2d}$, at the lattice $\Lambda$ (see Ref. [21]). These samples
are also a particular case of the average samples $\langle
T,\alpha_{\lambda}(Q_{m})\big{\rangle}_{\mathcal{H}\mathcal{S}}$ where the
average operator $Q_{m}$ is the rank-one operator $\widetilde{g}_{m}\otimes
g_{m}$; average sampling has been used previously in Refs. [6, 12].

The main aim here is the stable recovery of any $T\in V_{\bf S}^{2}$ from its
samples (1) by means of a sampling formula in $V_{\bf S}^{2}$ having the form

$T=\sum_{m=1}^{M}\sum_{\lambda\in\Lambda}s_{{}_{T,m}}(\lambda)\,\alpha_{\lambda}(H_{m})\quad\text{in
$\mathcal{H}\mathcal{S}$-norm}\,,$

for each $T\in V_{\bf S}^{2}$. The operators $H_{m}$, $m=1,2,\dots,M$, above
belong to $V_{\bf S}^{2}$ and satisfy that the sequence
$\\{\alpha_{\lambda}(H_{m})\\}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$ is a frame
for the Hilbert space $V_{\bf S}^{2}$.

For sampling in classical shift-invariant spaces see, for instance, Refs. [1,
13, 14] and references therein. See also Ref. [8] for the case where other
unitary representation of $\mathbb{R}$ on $L^{2}(\mathbb{R})$ is used instead
of the classical one given by translations. For the less known topic on
sampling operators, see Refs. [6, 12, 18, 20, 22, 23].

The used techniques in this work are those of the frame theory in a separable
Hilbert space. To be precise, the samples used along this paper will be
expressed as a discrete convolution system in the product Hilbert space
$\ell^{2}_{{}_{N}}(\Lambda):=\ell^{2}(\Lambda)\times\dots\times\ell^{2}(\Lambda)$
($N$ times), and then it will be used the close relationship between a
discrete convolution system and a sequence of translates in
$\ell^{2}_{{}_{N}}(\Lambda)$ (see, for instance, Ref. [15]). The other
involved tools are the Kohn-Nirenberg transform or the Weyl transform for
Hilbert-Schmidt operators: both are unitary operators from
$L^{2}(\mathbb{R}^{2d})$ onto $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ which
respect the translations in the sense that, if we denote any of them by
$\mathcal{L}$, we have $\mathcal{L}(T_{z}f)=\alpha_{z}(\mathcal{L}f)$ for
$f\in L^{2}(\mathbb{R}^{2d})$ and $z\in\mathbb{R}^{2d}$.

Now we briefly explain a practical motivation for considering the samples
defined in Eq. (1) for the elements in $V_{\bf S}^{2}$. It is a well-known
fact in mobile wireless channels that the relative location between
transmitter and receiver is varying with time and consequently the input-
output relation is modeled by a time-varying system $x\mapsto Hx$ that can be
expressed as the integral operator

$Hx(t)=\int_{\mathbb{R}^{d}}h_{t}(s)\,x(t-s)\,ds=\int_{\mathbb{R}^{d}}\sigma(t,\omega)\,\widehat{x}(\omega)\,{\rm
e}^{2\pi i\,\omega\cdot t}d\omega\,,$

where $\sigma(t,\omega)=\mathcal{F}(h_{t})(\omega)$, i.e., the Fourier
transform with respect to the last $d$ variables in $h(t,s):=h_{t}(s)$. In
this last formulation, operator $H$ becomes a pseudodifferential operator with
Kohn-Nirenberg symbol $\sigma$ (see, for instance, Refs. [16, 25]).

As it was pointed out in Ref. [18], in orthogonal frequency-division
multiplexing (OFDM) the digital information, i.e., a sequence of numbers
$\\{c_{\lambda}\\}$, $\lambda$ in the lattice $\Lambda=a\mathbb{Z}^{d}\times
b\mathbb{Z}^{d}$ $(a,b>0)$, is used as the coefficients of the input signal
$x(t)=\sum_{\mu\in\Lambda}c_{\mu}\,\pi(\lambda)g(t)$ of a time-varying system
$H$ producing the output $y(t)=Hx(t)$. Then, the sequence of numbers

$d_{\lambda}=\big{\langle}y,\pi(\lambda)\widetilde{g}\,\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\sum_{\mu\in\Lambda}c_{\mu}\,\big{\langle}H\pi(\mu)g,\pi(\lambda)\widetilde{g}\,\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\,,\quad\lambda\in\Lambda\,,$
(2)

is considered. The main task of the engineer is to recover the original data
$\\{c_{\lambda}\\}$ from the received data $\\{d_{\lambda}\\}$. The matrix
$A=[a_{\lambda,\mu}]$, where
$a_{\lambda,\mu}=\big{\langle}H\pi(\mu)g,\pi(\lambda)\widetilde{g}\,\big{\rangle}_{L^{2}(\mathbb{R}^{d})}$,
which appears in Eq. (2), involving $H$ and the time-frequency shifts of a
pair of fixed functions $g,\widetilde{g}\in L^{2}(\mathbb{R}^{d})$, is the so-
called channel matrix associated with $H$ and the functions $g,\widetilde{g}$
in $L^{2}(\mathbb{R}^{d})$. As it will be proved in Section 3.3 (see Eq. (8)
below), we have that

$\big{\langle}H\pi(\lambda)g,\pi(\lambda)\widetilde{g}\,\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\big{\langle}\alpha_{-\lambda}(H)g,\widetilde{g}\,\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\,,\quad\lambda\in\Lambda\,,$

i.e., the samples
$\big{\langle}\alpha_{-\lambda}(H)g,\widetilde{g}\,\big{\rangle}_{L^{2}(\mathbb{R}^{d})}$,
$\lambda\in\Lambda$, coincide with the diagonal entries of the channel matrix
associated with $H$ and windows $g,\widetilde{g}$. This is the reason to
consider the samples defined in Eq. (1) and to name them as the diagonal
channel samples of the operator $H$ with respect to the fixed functions
$g,\widetilde{g}\in L^{2}(\mathbb{R}^{d})$ and lattice $\Lambda$.

Besides, a simple class of operators $H$ describing time-varying systems, and
allowing to live in the Hilbert space setting, is given by the class of
Hilbert-Schmidt operators on $L^{2}(\mathbb{R}^{d})$. A Hilbert-Schmidt
operator $H$ on $L^{2}(\mathbb{R}^{d})$ is a compact operator on
$L^{2}(\mathbb{R}^{d})$ having the integral representation

$Hx(t)=\int_{\mathbb{R}^{d}}\kappa(t,s)\,x(s)ds=\int_{\mathbb{R}^{d}}\kappa(t,t-s)\,x(t-s)ds\,,$

with kernel $\kappa\in L^{2}(\mathbb{R}^{2d})$. Although only Hilbert-Schmidt
operators on $L^{2}(\mathbb{R}^{d})$ can be described as integral operators
with kernel in $L^{2}(\mathbb{R}^{2d})$, every bounded operator on
$L^{2}(\mathbb{R}^{d})$ can be uniquely described, via the Schwartz kernel
theorem, by a distributional kernel in $\mathcal{S}^{\prime}(\mathbb{R}^{2d})$
(see, for instance, Ref. [16]).

The paper is organized as follows: Section 2 introduces, for the sake of
completeness, some preliminaries needed in the sequel; they comprise Hilbert-
Schmidt operators and their Kohn-Nirenberg and Weyl transforms, the concept of
translation of an operator, and symplectic Fourier series. For the theory of
bases and frames in a Hilbert space we cite Ref. [3]. Section 3 contains the
main sampling results for the multiple generated subspace $V_{\bf S}^{2}$ of
$\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$. They rely on the expression of the
involved samples as the output of a bounded discrete convolution system
$\ell^{2}_{{}_{N}}(\Lambda)\rightarrow\ell^{2}_{{}_{M}}(\Lambda)$, and its
relationship with a frame of translates for $\ell^{2}_{{}_{N}}(\Lambda)$.

##  2 Some preliminaries

Next we briefly introduce some mathematical tools used throughout the work.
For the needed theory of bases and frames in a Hilbert space we merely make
reference to [3]; it mainly comprises Riesz sequences, dual Riesz bases and
frames and its duals in a separable Hilbert space. The results for discrete
convolution systems and their relationship with frames of translates in
$\ell^{2}_{{}_{N}}(\Lambda)$ can be found, for instance, in Ref. [15].

#### The Kohn-Nirenberg and Weyl transforms in the class of Hilbert-Schmidt
operators

The class of Hilbert-Schmidt operators in a Hilbert space,
$L^{2}(\mathbb{R}^{d})$ in our case, can be introduced by using the Schmidt
decomposition (singular value decomposition) of a compact operator on
$L^{2}(\mathbb{R}^{d})$ (see, for instance, Ref. [4]). Namely, for a compact
operator $S$ on $L^{2}(\mathbb{R}^{d})$ there exist two orthonormal sequences
$\\{x_{n}\\}_{n\in\mathbb{N}}$ and $\\{y_{n}\\}_{n\in\mathbb{N}}$ in
$L^{2}(\mathbb{R}^{d})$ and a bounded sequence of positive numbers
$\\{s_{n}(S)\\}_{n\in\mathbb{N}}$ (singular values of $S$) such that

$S=\sum_{n\in\mathbb{N}}s_{n}(S)\,x_{n}\otimes y_{n}\,,$

with convergence of the series in the operator norm. Here, $x_{n}\otimes
y_{n}$ denotes the rank-one operator defined by $\big{(}x_{n}\otimes
y_{n}\big{)}(e)=\big{\langle}e,y_{n}\big{\rangle}_{L^{2}}x_{n}$ for $e\in
L^{2}(\mathbb{R}^{d})$. For $1\leq p<\infty$ we define the Schatten-$p$ class
$\mathcal{T}^{p}$ by

$\mathcal{T}^{p}:=\big{\\{}S\,\text{ compact on
$L^{2}(\mathbb{R}^{d})$}\,\,:\,\,\\{s_{n}(S)\\}_{n\in\mathbb{N}}\in\ell^{p}(\mathbb{N})\big{\\}}\,.$

The Schatten-$p$ class $\mathcal{T}^{p}$ is a Banach space endowed with the
norm $\|S\|_{\mathcal{T}^{p}}^{p}=\sum_{n\in\mathbb{N}}s_{n}^{p}(S)$.

In particular, for $p=1$ we obtain the so-called trace class operators
$\mathcal{T}^{1}$. The trace defined by ${\rm
tr}(S)=\sum_{n\in\mathbb{N}}\langle Se_{n},e_{n}\rangle_{L^{2}}$ is a well-
defined bounded linear functional on $\mathcal{T}^{1}$, and independent of the
used orthonormal basis $\\{e_{n}\\}_{n\in\mathbb{N}}$ in
$L^{2}(\mathbb{R}^{d})$.

For $p=2$ we obtain the class of Hilbert-Schmidt operators
$\mathcal{H}\mathcal{S}(\mathbb{R}^{d}):=\mathcal{T}^{2}$. The space
$\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ endowed with the inner product
$\big{\langle}S,T\big{\rangle}_{\mathcal{H}\mathcal{S}}={\rm tr}(ST^{*})$
becomes a Hilbert space. For the norm of
$S\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ we have

$\|S\|_{\mathcal{H}\mathcal{S}}^{2}={\rm
tr}(SS^{*})=\sum_{n\in\mathbb{N}}\|S^{*}(e_{n})\|^{2}_{L^{2}}=\sum_{n\in\mathbb{N}}\|S(e_{n})\|^{2}_{L^{2}}=\sum_{n\in\mathbb{N}}s_{n}^{2}(S)\,.$

A Hilbert-Schmidt operator $S\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ can be
seen also as a compact operator on $L^{2}(\mathbb{R}^{d})$ defined for each
$f\in L^{2}(\mathbb{R}^{d})$ by

$Sf(t)=\int_{\mathbb{R}^{d}}\kappa_{{}_{S}}(t,x)f(x)dx\quad\text{ a.e.
$t\in\mathbb{R}^{d}$}\,,$

with kernel $\kappa_{{}_{S}}\in L^{2}(\mathbb{R}^{2d})$. Besides,
$\big{\langle}S,T\big{\rangle}_{\mathcal{H}\mathcal{S}}=\big{\langle}\kappa_{{}_{S}},\kappa_{{}_{T}}\big{\rangle}_{L^{2}(\mathbb{R}^{2d})}$
for $S,T\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$.

Now, we briefly introduce the Kohn-Nirenberg and Weyl transforms in
$L^{2}(\mathbb{R}^{2d})$, the setting where they will be used in this paper.
More information and details about these transforms, also valid in more
general settings, can be found in Refs. [7, 9, 16, 24, 26].

The Kohn-Nirenberg transform $L^{2}(\mathbb{R}^{2d})\ni\sigma\longmapsto
K_{\sigma}\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ is a unitary operator
where $K_{\sigma}:L^{2}(\mathbb{R}^{d})\rightarrow L^{2}(\mathbb{R}^{d})$ is
the Hilbert-Schmidt operator defined in weak sense by

$\big{\langle}K_{\sigma}\phi,\psi\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\big{\langle}\sigma,R(\psi,\phi)\big{\rangle}_{L^{2}(\mathbb{R}^{2d})}\,,\quad\phi,\psi\in
L^{2}(\mathbb{R}^{d})\,;$ (3)

here

$R(\psi,\phi)(x,\omega)=\psi(x)\,\overline{\widehat{\phi}(\omega)}\,{\rm
e}^{-2\pi i\,{x\cdot\omega}}\,,\quad(x,\omega)\in\mathbb{R}^{2d}\,,$

is the Rihaczek distribution of the functions $\psi,\phi\in
L^{2}(\mathbb{R}^{d})$ (see [16, Theorem 14.6.1]).

Thus, for each operator $S\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ there
exists a unique function $\sigma_{{}_{S}}\in L^{2}(\mathbb{R}^{2d})$, called
its Kohn-Nirenberg symbol, i.e. $S=K_{\sigma_{{}_{S}}}$, and such that

$\langle
S,T\rangle_{\mathcal{H}\mathcal{S}}=\langle\sigma_{{}_{S}},\sigma_{{}_{T}}\rangle_{L^{2}(\mathbb{R}^{2d})}\quad\text{for
each $S,T\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$}\,.$

The Weyl transform $L^{2}(\mathbb{R}^{2d})\ni f\longmapsto
L_{f}\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ is also a unitary operator
where $L_{f}:L^{2}(\mathbb{R}^{d})\rightarrow L^{2}(\mathbb{R}^{d})$ is the
Hilbert-Schmidt operator defined in weak sense by

$\big{\langle}L_{f}\phi,\psi\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\big{\langle}f,W(\psi,\phi)\big{\rangle}_{L^{2}(\mathbb{R}^{2d})}\,,\quad\phi,\psi\in
L^{2}(\mathbb{R}^{d})\,;$ (4)

here

$W(\psi,\phi)(x,\omega)=\int_{\mathbb{R}^{d}}\psi\big{(}x+\frac{t}{2}\big{)}\,\overline{\phi\big{(}x-\frac{t}{2}\big{)}}\,{\rm
e}^{-2\pi i\,{\omega\cdot t}}dt\,,\quad(x,\omega)\in\mathbb{R}^{2d}\,,$

is the cross-Wigner distribution of the functions $\psi,\phi\in
L^{2}(\mathbb{R}^{d})$ (see Ref. [16, Theorem 14.6.1]).

Thus, for each operator $S\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ there
exists a unique function $a_{{}_{S}}\in L^{2}(\mathbb{R}^{2d})$, called its
Weyl symbol, i.e. $S=L_{a_{{}_{S}}}$, and such that

$\langle S,T\rangle_{\mathcal{H}\mathcal{S}}=\langle
a_{{}_{S}},a_{{}_{T}}\rangle_{L^{2}(\mathbb{R}^{2d})}\quad\text{for each
$S,T\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$}\,.$

If $a_{{}_{S}}$ denotes the Weyl symbol of $S$, its Kohn-Nirenberg symbol
$\sigma_{{}_{S}}$ is given by $Ua_{{}_{S}}$ where $U$ is the unitary operator
on $L^{2}(\mathbb{R}^{2d})$ such that $\widehat{Ua_{{}_{S}}}(\xi,u)={\rm
e}^{\pi iu\cdot\xi}\,\widehat{a}_{{}_{S}}(\xi,u)$, $(\xi,u)\in\mathbb{R}^{2d}$
(see the details in Ref. [16]).

The Kohn-Nirenberg (or Weyl) transform can be defined for $\sigma$ (or $f$) in
$\mathcal{S}^{\prime}(\mathbb{R}^{2d})$, i.e., for tempered distributions by
using the dualities
$\big{(}\mathcal{S}(\mathbb{R}^{d}),\mathcal{S}^{\prime}(\mathbb{R}^{d})\big{)}$
and
$\big{(}\mathcal{S}(\mathbb{R}^{2d}),\mathcal{S}^{\prime}(\mathbb{R}^{2d})\big{)}$
in Eq. (3) (or Eq. (4)); see, for instance, Refs. [16, 24].

#### Translation of operators

For $z=(x,\omega)\in\mathbb{R}^{2d}$, the time-frequency shift operator
$\pi(z):L^{2}(\mathbb{R}^{d})\rightarrow L^{2}(\mathbb{R}^{d})$ is defined as

$\pi(z)\varphi(t)={\rm e}^{2\pi i\omega\cdot t}\varphi(t-x)\quad\text{for
$\varphi\in L^{2}(\mathbb{R}^{d})$}\,.$

It is used to define the short-time Fourier transform (Gabor transform)
$V_{\psi}\varphi$ of $\varphi$ with window $\psi$, both in
$L^{2}(\mathbb{R}^{d})$, by

$V_{\psi}\varphi(z)=\big{\langle}\varphi,\pi(z)\psi\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\,,\quad
z\in\mathbb{R}^{2d}\,.$

Its adjoint operator is $\pi(z)^{*}={\rm e}^{-2\pi ix\cdot\omega}\,\pi(-z)$
for $z=(x,\omega)\in\mathbb{R}^{2d}$. By using conjugation with $\pi(z)$ one
can define the translation by $z\in\mathbb{R}^{2d}$ of an operator
$S\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$. Namely,

$\alpha_{z}(S):=\pi(z)\,S\,\pi(z)^{*}\,,\quad z\in\mathbb{R}^{2d}\,.$

For instance, for $\varphi,\psi\in L^{2}(\mathbb{R}^{d})$ we get
$\alpha_{z}(\varphi\otimes\psi)=[\pi(z)\varphi]\otimes[\pi(z)\psi]$,
$z\in\mathbb{R}^{2d}$.

Since $\alpha_{z}$ defines a unitary operator on
$\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$,
$\alpha_{z}\alpha_{z^{\prime}}=\alpha_{z+z^{\prime}}$ for
$z,z^{\prime}\in\mathbb{R}^{2d}$, and the map $z\mapsto\alpha_{z}(S)$ is
continuous for each $S\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ we have that
$\big{\\{}\alpha_{z}\big{\\}}_{z\in\mathbb{R}^{2d}}$ is a unitary
representation of the group $\mathbb{R}^{2d}$ on the Hilbert space
$\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$. More properties and applications can
be found, for instance, in Refs. [21, 24, 26].

#### Symplectic Fourier series

Let $\Lambda$ be a full rank lattice in $\mathbb{R}^{2d}$, i.e.,
$\Lambda=A\mathbb{Z}^{2d}$ with $A\in GL(2d,\mathbb{R})$ and volume
$|\Lambda|=\det A$. Its dual group $\widehat{\Lambda}$ is identified with
$\mathbb{R}^{2d}/\Lambda^{\circ}$, where $\Lambda^{\circ}$ is the annihilator
group

$\Lambda^{\circ}=\big{\\{}\lambda^{\circ}\in\mathbb{R}^{2d}\,\,:\,\,{\rm
e}^{2\pi i\,\sigma(\lambda^{\circ},\lambda)}=1\,\,\text{ for all
$\lambda\in\Lambda$}\big{\\}}\,,$

where $\sigma$ denotes here the standard symplectic form
$\sigma(z,z^{\prime})=\omega\cdot x^{\prime}-\omega^{\prime}\cdot x$ for
$z=(x,\omega)$ and $z^{\prime}=(x^{\prime},\omega^{\prime})$ in
$\mathbb{R}^{2d}$. Notice that, since $\Lambda$ is discrete its dual group
$\widehat{\Lambda}$ is compact. The group $\Lambda^{\circ}$ is itself a
lattice: the so-called adjoint lattice of $\Lambda$. The symplectic characters
$\chi_{z}(z^{\prime}):={\rm e}^{2\pi i\,\sigma(z,z^{\prime})}$ are the natural
way of identifying the group $\mathbb{R}^{2d}$ with its dual group via the
bijection $z\mapsto\chi_{z}$.

The Fourier transform of $c\in\ell^{1}(\Lambda)$ is the symplectic Fourier
series

$\mathcal{F}_{s}^{\Lambda}(c)(\dot{z}):=\sum_{\lambda\in\Lambda}c(\lambda)\,{\rm
e}^{2\pi
i\,\sigma(\lambda,z)}\,,\quad\dot{z}\in\mathbb{R}^{2d}/\Lambda^{\circ}\,,$

where $\dot{z}$ denotes the image of $z$ under the natural quotient map
$\mathbb{R}^{2d}\rightarrow\mathbb{R}^{2d}/\Lambda^{\circ}$.

Since $\mathcal{F}_{s}^{\Lambda}$ is a Fourier transform it extends to a
unitary mapping $\mathcal{F}_{s}^{\Lambda}:\ell^{2}(\Lambda)\rightarrow
L^{2}(\,\widehat{\Lambda}\,)$. It satisfies
$\mathcal{F}_{s}^{\Lambda}(c\ast_{\Lambda}d)=\mathcal{F}_{s}^{\Lambda}(c)\,\mathcal{F}_{s}^{\Lambda}(d)$
for $c\in\ell^{1}(\Lambda)$ and $d\in\ell^{2}(\Lambda)$. Moreover, if
$c,d\in\ell^{2}(\Lambda)$ with $c\ast_{\Lambda}d\in\ell^{2}(\Lambda)$, then
$\mathcal{F}_{s}^{\Lambda}(c\ast_{\Lambda}d)=\mathcal{F}_{s}^{\Lambda}(c)\,\mathcal{F}_{s}^{\Lambda}(d)$.
As usual, the convolution $\ast_{\Lambda}$ of two sequences $c,d$ is defined
by

$\big{(}c\ast_{\Lambda}d\big{)}(\lambda)=\sum_{\mu\in\Lambda}c(\mu)\,d(\lambda-\mu),\quad\lambda\in\Lambda\,.$

For more details, see, for instance, Refs. [5, 10, 11, 24].

##  3 Sampling in the case of multiple generators

For a fixed set ${\bf
S}=\\{S_{1},S_{2},\dots,S_{N}\\}\subset\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$,
we are interested that the sequence of translates
$\\{\alpha_{\lambda}(S_{n})\\}_{\lambda\in\Lambda;\,n=1,2,\dots,N}$ forms a
Riesz sequence for $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ where
$\Lambda\subset\mathbb{R}^{2d}$ is a full rank lattice with dual group
$\widehat{\Lambda}$.

###  3.1 Riesz sequences of translated operators in
$\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$

As it was said before, the Weyl transform $f\mapsto L_{f}$ is a unitary
operator
$L^{2}(\mathbb{R}^{2d})\rightarrow\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$
which respects translations in the sense that

$L_{T_{z}f}=\alpha_{z}(L_{f})\quad\text{for $f\in L^{2}(\mathbb{R}^{2d})$ and
$z\in\mathbb{R}^{2d}$}\,.$

These two properties are very important throughtout this work. In particular,
as it was pointed out in Refs. [6, 24], for fixed
$S\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ with Weyl symbol $a_{{}_{S}}\in
L^{2}(\mathbb{R}^{2d})$ and lattice $\Lambda$ in $\mathbb{R}^{2d}$, the
sequence $\\{\alpha_{\lambda}(S)\\}_{\lambda\in\Lambda}$ is a Riesz sequence
in $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$, i.e., a Riesz basis for
$V_{S}^{2}:=\overline{\operatorname{span}}_{\mathcal{H}\mathcal{S}}\big{\\{}\alpha_{\lambda}(S)\big{\\}}_{\lambda\in\Lambda}$,
if and only if the sequence
$\\{T_{\lambda}(a_{{}_{S}})\\}_{\lambda\in\Lambda}$ is a Riesz sequence in
$L^{2}(\mathbb{R}^{2d})$, i.e., a Riesz basis for the shift-invariant subspace
$V_{a_{S}}^{2}$ in $L^{2}(\mathbb{R}^{2d})$ generated by $a_{{}_{S}}$.

A necessary and sufficient condition for
$\\{\alpha_{\lambda}(S)\\}_{\lambda\in\Lambda}$ to be a Riesz sequence in
$\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ is given in Ref. [24]. There, it is
assumed that $S\in\mathcal{B}$, a Banach space of continuous operators with
Weyl symbol $a_{{}_{S}}$ in the Feichtinger’s algebra
$\mathcal{S}_{0}(\mathbb{R}^{2d})$; in essence, $\mathcal{B}$ consists of
trace class operators on $L^{2}(\mathbb{R}^{d})$ with a norm-continuous
inclusion $\iota:\mathcal{B}\hookrightarrow\mathcal{T}^{1}$ (see the details
in Refs. [17, 24]).

Recall that the Feichtinger’s algebra $\mathcal{S}_{0}(\mathbb{R}^{d})$ is the
space of all tempered distributions $\psi$ in $\mathbb{R}^{d}$ such that

$\|\psi\|_{\mathcal{S}_{0}}:=\int_{\mathbb{R}^{2d}}|V_{\varphi_{0}}\psi(z)|dz<\infty\,,$

where $\varphi_{0}$ denotes the $L^{2}$-normalized gaussian
$\varphi_{0}(x)=2^{d/4}{\rm e}^{-\pi x\cdot x}$ for $x\in\mathbb{R}^{d}$. With
this norm, $\mathcal{S}_{0}(\mathbb{R}^{d})$ is a Banach space of continuous
functions and an algebra under multiplication and convolution; see the details
in Refs. [16, 19, 24].

######  Theorem 1.

([24, Theorem 6.1]) Let $\Lambda$ be a lattice and $S\in\mathcal{B}$. The
sequence $\\{\alpha_{\lambda}(S)\\}_{\lambda\in\Lambda}$ is a Riesz sequence
in $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ if and only if the function

$P_{\Lambda^{\circ}}\big{(}|\mathcal{F}_{W}(S)|^{2}\big{)}(\dot{z}):=\frac{1}{|\Lambda|}\sum_{\lambda^{\circ}\in\Lambda^{\circ}}|\mathcal{F}_{W}(S)(z+\lambda^{\circ})|^{2}\,,\quad
z\in\mathbb{R}^{2d}\,,$

has no zeros in $\widehat{\Lambda}$.

It involves the periodization operator $P_{\Lambda^{\circ}}$ in
$\Lambda^{\circ}$ and the Fourier-Wigner transform $\mathcal{F}_{W}$ of an
operator $S$. In this case, we have that
$\mathcal{F}_{W}(S)=\mathcal{F}_{s}(a_{{}_{S}})$, where $\mathcal{F}_{s}$
denotes the symplectic Fourier transform of $a_{{}_{S}}$ defined by

$\mathcal{F}_{s}(a_{{}_{S}})(z):=\int_{\mathbb{R}^{2d}}a_{{}_{S}}(z^{\prime})\,{\rm
e}^{-2\pi i\,\sigma(z,z^{\prime})}dz^{\prime}\,,\quad z\in\mathbb{R}^{2d}\,,$

where $\sigma$ denotes here the standard symplectic form in $\mathbb{R}^{2d}$.
The Fourier-Wigner transform of an operator $S$ is defined as the function

$\mathcal{F}_{W}(S)(z):={\rm e}^{-\pi i\,x\cdot\omega}\,{\rm
tr}[\pi(-z)S]\,,\quad z=(x,\omega)\in\mathbb{R}^{2d}\,.$

See the details in Ref. [24]. A similar result to that in the above theorem
for a rank-one operator $S=\psi\otimes\phi$, where $\psi,\phi\in
L^{2}(\mathbb{R}^{d})$, can be found in Refs. [2, 6].

In case $\\{\alpha_{\lambda}(S)\\}_{\lambda\in\Lambda}$ is a Riesz sequence
for $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$, the operator $S$ is the
generator of the $\Lambda$-shift-invariant subspace $V_{S}^{2}$ which can be
described by

$V_{S}^{2}:=\overline{\operatorname{span}}_{\mathcal{H}\mathcal{S}}\big{\\{}\alpha_{\lambda}(S)\big{\\}}_{\lambda\in\Lambda}=\Big{\\{}\sum_{\lambda\in\Lambda}c(\lambda)\,\alpha_{\lambda}(S)\,\,:\,\,\\{c(\lambda)\\}_{\lambda\in\Lambda}\in\ell^{2}(\Lambda)\Big{\\}}\,.$

Observe that operators in $V_{S}^{2}$ are nothing but Gabor multipliers in
case $S=\varphi\otimes\psi$. Indeed, for $\eta\in L^{2}(\mathbb{R}^{d})$ we
have

$\sum_{\lambda\in\Lambda}c(\lambda)\,\alpha_{\lambda}(S)(\eta)=\sum_{\lambda\in\Lambda}c(\lambda)\,\big{(}\pi(\lambda)\varphi\otimes\pi(\lambda)\psi\big{)}(\eta)=\sum_{\lambda\in\Lambda}c(\lambda)\,V_{\psi}\,\eta(\lambda)\pi(\lambda)\varphi\,,$

that is,
$\sum_{\lambda\in\Lambda}c(\lambda)\,\alpha_{\lambda}(S)=\mathcal{G}_{\mathbf{c}}^{\psi,\varphi}$,
the Gabor multiplier with windows $\psi,\varphi$ and mask $\mathbf{c}$ in
$\ell^{2}(\Lambda)$ used in time-frequency analysis (see, for instance, Ref.
[24]).

Analogously, a necessary and sufficient condition can be obtained for the
multiply generated case. Indeed, let ${\bf S}=\\{S_{1},S_{2},\dots,S_{N}\\}$
be a fixed subset of $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ and let
$\Lambda$ be a lattice in $\mathbb{R}^{2d}$. We are searching for a necessary
and sufficient condition such that
$\\{\alpha_{\lambda}(S_{n})\\}_{\lambda\in\Lambda;\,n=1,2,\dots,N}$ is a Riesz
sequence for $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$, i.e., a Riesz basis for
the closed subspace

$V_{\bf
S}^{2}:=\overline{\operatorname{span}}_{\mathcal{H}\mathcal{S}}\big{\\{}\alpha_{\lambda}(S_{n})\big{\\}}_{\lambda\in\Lambda;\,n=1,2,\dots,N}\subset\mathcal{H}\mathcal{S}(\mathbb{R}^{d})\,.$

For the multiply generated case we have the following result:

######  Theorem 2.

Let $\Lambda$ be a lattice and $S_{n}\in\mathcal{B}$, $n=1,2,\dots,N$. Then,
$\\{\alpha_{\lambda}(S_{n})\\}_{\lambda\in\Lambda;\,n=1,2,\dots,N}$ is a Riesz
sequence for $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ if and only if there
exist two constants $0<m\leq M$ such that

$m\,\mathbb{I}_{{}_{N}}\leq G_{\bf S}^{W}(z)\leq
M\,\mathbb{I}_{{}_{N}}\quad\text{for any $z\in\mathbb{R}^{2d}$}\,,$

where $G_{\bf S}^{W}(z)$ denotes the $N\times N$ matrix-valued function

$G_{\bf
S}^{W}(z):=\sum_{\lambda^{\circ}\in\Lambda^{\circ}}\mathcal{F}_{W}({\bf
S})(z+\lambda^{\circ})\,\overline{\mathcal{F}_{W}({\bf
S})(z+\lambda^{\circ})}^{\top}\,,\quad z\in\mathbb{R}^{2d}\,,$

and $\mathcal{F}_{W}({\bf
S})=\big{(}\mathcal{F}_{W}(S_{1}),\mathcal{F}_{W}(S_{2}),\dots,\mathcal{F}_{W}(S_{N})\big{)}^{\top}$.

###### Proof.

As indicated above, it will be a Riesz sequence in
$\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ if and only if the sequence
$\\{T_{\lambda}(a_{{}_{S_{n}}})\\}_{\lambda\in\Lambda;\,n=1,2,\dots,N}$ is a
Riesz sequence in $L^{2}(\mathbb{R}^{2d})$. To this end, we introduce the
$N\times N$ matrix-valued function

$G_{\bf
S}^{\sigma}(z):=\sum_{\lambda^{\circ}\in\Lambda^{\circ}}\mathcal{F}_{s}(a_{{}_{\bf
S}})(z+\lambda^{\circ})\,\overline{\mathcal{F}_{s}(a_{{}_{\bf
S}})(z+\lambda^{\circ})}^{\top}\,,\quad z\in\mathbb{R}^{2d}\,,$

where $\mathcal{F}_{s}(a_{{}_{\bf
S}})=\big{(}\mathcal{F}_{s}(a_{{}_{S_{1}}}),\mathcal{F}_{s}(a_{{}_{S_{2}}}),\dots,\mathcal{F}_{s}(a_{{}_{S_{N}}})\big{)}^{\top}$.
It is known (see, for instance, Ref. [1]) that the sequence
$\\{T_{\lambda}(a_{{}_{S_{n}}})\\}_{\lambda\in\Lambda;\,n=1,2,\dots,N}$ is a
Riesz sequence in $L^{2}(\mathbb{R}^{2d})$ if and only if there exist two
constants $0<m\leq M$ such that $m\,\mathbb{I}_{{}_{N}}\leq G_{\bf
S}^{\sigma}(z)\leq M\,\mathbb{I}_{{}_{N}}$, a.e. $z\in\mathbb{R}^{2d}$, where
$\mathbb{I}_{{}_{N}}$ denotes the $N\times N$ identity matrix. Assuming as
before that $S_{n}\in\mathcal{B}$, $n=1,2,\dots,N$, the functions
$\mathcal{F}_{s}(a_{{}_{S_{n}}})$ are continuous and
$\mathcal{F}_{W}(S_{n})=\mathcal{F}_{s}(a_{{}_{S_{n}}})$ for $n=1,2,\dots,N$.
Hence, the above necessary and sufficient condition can be expressed in terms
of the hermitian matrix $G_{\bf S}^{W}(z)$ as in the statement of the theorem.
∎

In this case, ${\bf S}=\\{S_{1},S_{2},\dots,S_{N}\\}$ is a set of generators
for the $\Lambda$-shift-invariant subspace $V_{\bf
S}^{2}:=\overline{\operatorname{span}}_{\mathcal{H}\mathcal{S}}\big{\\{}\alpha_{\lambda}(S_{n})\big{\\}}_{\lambda\in\Lambda;\,n=1,2,\dots,N}$
which can be described by

$V_{\bf
S}^{2}=\Big{\\{}\sum_{n=1}^{N}\sum_{\lambda\in\Lambda}c_{n}(\lambda)\,\alpha_{\lambda}(S_{n})\,\,:\,\,\\{c_{n}(\lambda)\\}_{\lambda\in\Lambda}\in\ell^{2}(\Lambda)\,,\,n=1,2,\dots,N\Big{\\}}\,.$

###  3.2 The isomorphism $\mathcal{T}_{\bf S}$

Our sampling results rely on the following isomorphism $\mathcal{T}_{\bf S}$
which involves the spaces $\ell^{2}_{{}_{N}}(\Lambda)$, the shift-invariant
subspace $V_{\sigma_{{}_{\bf S}}}^{2}$ in $L^{2}(\mathbb{R}^{2d})$ generated
by the Kohn-Nirenberg symbols $\sigma_{{}_{S_{n}}}$ of $S_{n}$,
$n=1,2,\dots,N$, and the $\Lambda$-shift-invariant subspace $V_{\bf S}^{2}$.
Namely,

$\begin{array}[c]{cccccc}\mathcal{T}_{\bf
S}:&\ell^{2}_{{}_{N}}(\Lambda)&\longrightarrow&V_{\sigma_{\bf S}}^{2}\subset
L^{2}(\mathbb{R}^{2d})&\longrightarrow&V_{\bf
S}^{2}\subset\mathcal{H}\mathcal{S}(\mathbb{R}^{d})\phantom{\dfrac{a}{b}}\\\
&(c_{1},c_{2},\dots,c_{N})^{\top}&\longmapsto&\displaystyle{\sum_{n=1}^{N}\sum_{\lambda\in\Lambda}c_{n}(\lambda)\,T_{\lambda}\sigma_{{}_{S_{n}}}}&\longmapsto&\displaystyle{\sum_{n=1}^{N}\sum_{\lambda\in\Lambda}c_{n}(\lambda)\,\alpha_{\lambda}(S_{n})}\,.\end{array}$
(5)

The isomorphism $\mathcal{T}_{\bf S}$ is the composition of the isomorphism
$\mathcal{T}_{\sigma_{\bf S}}:\ell^{2}_{{}_{N}}(\Lambda)\rightarrow
V_{\sigma_{\bf S}}^{2}$ which maps the standard orthonormal basis
$\\{\boldsymbol{\delta}_{\lambda}\\}_{\lambda\in\Lambda}$ for
$\ell^{2}_{{}_{N}}(\Lambda)$ onto the Riesz basis
$\\{T_{\lambda}\sigma_{{}_{S_{n}}}\\}_{\lambda\in\Lambda;\,n=1,2,\dots,N}$ for
$V_{\sigma_{\bf S}}^{2}$, and the Kohn-Nirenberg transform transform between
$V_{\sigma_{\bf S}}^{2}$ and $V_{\bf S}^{2}$.

Recall that the Kohn-Nirenberg transform $L^{2}(\mathbb{R}^{2d})\ni f\mapsto
K_{f}\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ is a unitary operator which
respects translations in the sense that $K_{T_{z}f}=\alpha_{z}(K_{f})$ for
$f\in L^{2}(\mathbb{R}^{2d})$ and $z\in\mathbb{R}^{2d}$. See, for instance,
Ref. [7, 16].

###  3.3 An expression for the samples

For each $T=\sum_{n=1}^{N}\sum_{\mu\in\Lambda}c_{n}(\mu)\,\alpha_{\mu}(S_{n})$
in $V_{\bf S}^{2}$ we define a set of diagonal channel samples as

$\mathbf{s}_{{}_{T}}(\lambda):=\big{(}\langle\alpha_{-\lambda}(T)g_{1},\widetilde{g}_{1}\rangle,\langle\alpha_{-\lambda}(T)g_{2},\widetilde{g}_{2}\rangle,\dots,\langle\alpha_{-\lambda}(T)g_{{}_{M}},\widetilde{g}_{{}_{M}}\rangle\big{)}^{\top}\,,\quad\lambda\in\Lambda\,,$
(6)

where $g_{m},\widetilde{g}_{m}$, $m=1,2,\dots,M$, denote $2M$ fixed functions
in $L^{2}(\mathbb{R}^{d})$. For $m=1,2,\dots,M$ the above samples can be
expressed by

$\begin{split}s_{{}_{T,m}}(\lambda):=&\big{\langle}\alpha_{-\lambda}(T)g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\big{\langle}\sum_{n=1}^{N}\sum_{\mu\in\Lambda}c_{n}(\mu)\alpha_{\mu-\lambda}(S_{n})g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\\\
=&\sum_{n=1}^{N}\sum_{\mu\in\Lambda}c_{n}(\mu)\big{\langle}\alpha_{\mu-\lambda}(S_{n})g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\sum_{n=1}^{N}\big{(}a_{m,n}\ast_{\Lambda}c_{n}\big{)}(\lambda)\,,\quad\lambda\in\Lambda\,,\end{split}$
(7)

where
$a_{m,n}(\mu):=\big{\langle}\alpha_{-\mu}(S_{n})g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}$,
$\mu\in\Lambda$. Observe that $a_{m,n}(\lambda)$, $\lambda\in\Lambda$, are
precisely the samples $\mathbf{s}_{{}_{S_{n}}}(\lambda)$, $\lambda\in\Lambda$,
of the generator $S_{n}$.

######  Lemma 3.

Concerning the samples defined in Eq. (7) we have:

  1. 1.

For $m=1,2,\dots,M$ these samples can be written as

$\big{\langle}\alpha_{-\lambda}(T)g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\big{\langle}T\pi(\lambda)g_{m},\pi(\lambda)\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\big{\langle}T,\alpha_{\lambda}(\widetilde{g}_{m}\otimes
g_{m})\big{\rangle}_{{}_{\mathcal{H}\mathcal{S}}}\,,\,\,\lambda\in\Lambda\,.$
(8)

  2. 2.

The sequences $\big{\\{}a_{m,n}(\lambda)\big{\\}}_{\lambda\in\Lambda}$
appearing in Eq. (7) belong to $\ell^{2}(\Lambda)$ for $m=1,2,\dots,M$ and
$n=1,2,\dots,N$.

###### Proof.

For the first equality in (8) we have that

$\begin{split}s_{{}_{T,m}}(\lambda)&=\big{\langle}\alpha_{-\lambda}(T)g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\big{\langle}\pi(-\lambda)T\pi(-\lambda)^{*}g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\\\
&=\big{\langle}\sigma_{{}_{T}},R\big{(}\pi(-\lambda)^{*}\widetilde{g}_{m},\pi(-\lambda)^{*}g_{m}\big{)}\big{\rangle}_{L^{2}(\mathbb{R}^{2d})}\,,\quad\lambda\in\Lambda\,.\end{split}$

On the other hand, it is easy to check that for the Rihaczek distribution one
gets

$R\big{(}\pi(-\lambda)^{*}\widetilde{g}_{m},\pi(-\lambda)^{*}g_{m}\big{)}(z)=R\big{(}\pi(\lambda)\widetilde{g}_{m},\pi(\lambda)g_{m}\big{)}(z)\,,\quad
z\in\mathbb{R}^{2d}\,.$

Hence, for each $\lambda\in\Lambda$ we obtain

$\big{\langle}\alpha_{-\lambda}(T)g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\big{\langle}\sigma_{{}_{T}},R\big{(}\pi(\lambda)\widetilde{g}_{m},\pi(\lambda)g_{m}\big{)}\big{\rangle}_{L^{2}(\mathbb{R}^{2d})}=\big{\langle}T\pi(\lambda)g_{m},\pi(\lambda)\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\,.$

For the second equality we get

$\begin{split}\big{\langle}T,\alpha_{\lambda}(\widetilde{g}_{m}\otimes
g_{m})\big{\rangle}_{{}_{\mathcal{H}\mathcal{S}}}&=\big{\langle}T,\pi(\lambda)\widetilde{g}_{m}\otimes\pi(\lambda)g_{m}\big{\rangle}_{{}_{\mathcal{H}\mathcal{S}}}=\big{\langle}\sigma_{T},\sigma_{\pi(\lambda)\widetilde{g}_{m}\otimes\pi(\lambda)g_{m}}\big{\rangle}_{L^{2}(\mathbb{R}^{2d})}\\\
&=\big{\langle}\sigma_{T},R(\pi(\lambda)\widetilde{g}_{m},\pi(\lambda)g_{m})\big{\rangle}_{L^{2}(\mathbb{R}^{2d})}=\big{\langle}T\pi(\lambda)g_{m},\pi(\lambda)\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\,.\end{split}$

We have used that the Kohn-Nirenberg symbol of
$\pi(\lambda)\widetilde{g}_{m}\otimes\pi(\lambda)g_{m}$ coincides with the
Rihaczek distribution of the pair of functions $\pi(\lambda)\widetilde{g}_{m}$
and $\pi(\lambda)g_{m}$ in $L^{2}(\mathbb{R}^{d})$.

In particular we have proved that

$a_{m,n}(\lambda)=\big{\langle}\alpha_{-\lambda}(S_{n})g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\big{\langle}S_{n},\alpha_{\lambda}(\widetilde{g}_{m}\otimes
g_{m})\big{\rangle}_{{}_{\mathcal{H}\mathcal{S}}}=\big{\langle}\alpha_{-\lambda}(S_{n}),\widetilde{g}_{m}\otimes
g_{m}\big{\rangle}_{{}_{\mathcal{H}\mathcal{S}}}\,,\quad\lambda\in\Lambda\,.$

Since $\\{\alpha_{\lambda}(S_{n})\\}_{\lambda\in\Lambda;\,n=1,2,\dots,N}$ is a
Riesz sequence for $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$, it is in
particular a Bessel sequence in $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$.
Hence, the sequences
$\big{\\{}\big{\langle}\alpha_{-\lambda}(S_{n}),\widetilde{g}_{m}\otimes
g_{m}\big{\rangle}_{{}_{\mathcal{H}\mathcal{S}}}\big{\\}}_{\lambda\in\Lambda}$
belongs to $\ell^{2}(\Lambda)$ for $m=1,2,\dots,M$ and $n=1,2,\dots,N$. ∎

Once we have that $a_{m,n}\in\ell^{2}(\Lambda)$ for each $m=1,2,\dots,M$ and
$n=1,2,\dots,N$, and denoting $A=[a_{m,n}]$ the corresponding $M\times N$
matrix with entries in $\ell^{2}(\Lambda)$, the sampling process in (6) is
described by means of the discrete convolution system

$T=\sum_{n=1}^{N}\sum_{\lambda\in\Lambda}c_{n}(\lambda)\,\alpha_{\lambda}(S_{n})\in
V_{\bf
S}^{2}\longmapsto\mathbf{s}_{{}_{T}}(\lambda)=\big{(}A\ast_{\Lambda}\mathbf{c}\big{)}(\lambda)=\sum_{\mu\in\Lambda}A(\lambda-\mu)\,\mathbf{c}(\mu),\quad\lambda\in\Lambda\,,$

where
$\mathbf{c}=(c_{1},c_{2},\dots,c_{N})^{\top}\in\ell^{2}_{{}_{N}}(\Lambda):=\ell^{2}(\Lambda)\times\dots\times\ell^{2}(\Lambda)$
($N$ times). Note that the $m$-th entry of $A\ast_{\Lambda}\mathbf{c}$ is
$\sum_{n=1}^{N}(a_{m,n}\ast_{\Lambda}c_{n})$.

First of all, the mapping
$\mathcal{A}:\ell^{2}_{{}_{N}}(\Lambda)\rightarrow\ell^{2}_{{}_{M}}(\Lambda)$
which maps $\mathbf{c}\mapsto A\ast_{\Lambda}\mathbf{c}$ is a well-defined
bounded operator if and only if the $M\times N$ matrix-valued function
$\widehat{A}(\xi):=\big{[}\mathcal{F}_{s}^{\Lambda}(a_{m,n})(\xi)\big{]}$,
a.e. $\xi\in\widehat{\Lambda}$, has entries in
$L^{\infty}(\widehat{\Lambda})$. The needed results on discrete convolution
systems
$\mathcal{A}:\ell^{2}_{{}_{N}}(\Lambda)\rightarrow\ell^{2}_{{}_{M}}(\Lambda)$,
and their relationship with frames of translates in
$\ell^{2}_{{}_{N}}(\Lambda)$ can be found in Ref. [15]. Notice that the $m$-th
component of $A\ast_{\Lambda}\mathbf{c}$ is

$[A\ast\mathbf{c}]_{m}(\lambda)=\sum_{n=1}^{N}(a_{m,n}\ast_{{}_{\Lambda}}c_{n})(\lambda)=\big{\langle}\mathbf{c},T_{\lambda}\,\mathbf{a}^{*}_{m}\big{\rangle}_{\ell^{2}_{{}_{N}}(\Lambda)}\,,\quad\lambda\in\Lambda\,,$

where $a^{*}_{m,n}$ denotes the involution
$a^{*}_{m,n}(\lambda):=\overline{a_{m,n}(-\lambda)}$, $\lambda\in\Lambda$. As
a consequence, the operator $\mathcal{A}$ is the analysis operator of the
sequence
$\big{\\{}T_{\lambda}\mathbf{a}^{*}_{m}\big{\\}}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$
in $\ell^{2}_{{}_{N}}(\Lambda)$. Since the sequence
$\big{\\{}T_{\lambda}\,\mathbf{a}^{*}_{m}\big{\\}}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$
is a frame for $\ell^{2}_{{}_{N}}(\Lambda)$ if and only if its bounded
analysis operator is injective with a closed range (see Ref. [3]), it will be
a frame for $\ell^{2}_{N}(\Lambda)$ if and only if

$0<\alpha_{A}:=\operatorname*{ess\,inf}_{\xi\in\widehat{\Lambda}}\lambda_{\text{min}}[\,\widehat{A}(\xi)^{*}\widehat{A}(\xi)]\leq\beta_{A}:=\operatorname*{ess\,sup}_{\xi\in\widehat{\Lambda}}\lambda_{\text{max}}[\widehat{A}(\xi)^{*}\widehat{A}(\xi)]<+\infty\,,$
(9)

where $\lambda_{\text{min}}$ (respectively, $\lambda_{\text{max}}$) denotes
the smallest (respectively, the largest) eigenvalue of the positive
semidefinite matrix $\widehat{A}(\xi)^{*}\widehat{A}(\xi)$ (see Ref. [15]).

Concerning the duals of
$\big{\\{}T_{\lambda}\,\mathbf{a}^{*}_{m}\big{\\}}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$
having its same structure, consider two matrices
$\widehat{A}\in\mathcal{M}_{{}_{M\times N}}(L^{\infty}(\widehat{\Lambda}))$
and $\widehat{B}\in\mathcal{M}_{{}_{N\times
M}}(L^{\infty}(\widehat{\Lambda}))$, and let $\mathbf{b}_{m}$ denote the
$m$-th column of the matrix $B$ associated to $\widehat{B}$. Then, the
sequences
$\big{\\{}T_{\lambda}\,\mathbf{a}^{*}_{m}\big{\\}}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$
and
$\big{\\{}T_{\lambda}\,\mathbf{b}_{m}\big{\\}}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$
form a pair of dual frames for $\ell^{2}_{{}_{N}}(\Lambda)$ if and only if
$\widehat{B}(\xi)\,\widehat{A}(\xi)=\mathbb{I}_{{}_{N}}$, a.e.
$\xi\in\widehat{\Lambda}$; equivalently, if and only if
$\mathcal{B}\,\mathcal{A}=\mathcal{I}_{\ell^{2}_{N}(\Lambda)}$, i.e., the
convolution system $\mathcal{B}$ with matrix $B$ is a left-inverse of the
convolution system $\mathcal{A}$ with matrix $A$. Thus, we have the frame
expansion

$\mathbf{c}=\sum_{m=1}^{M}\sum_{\lambda\in\Lambda}\big{\langle}\mathbf{c},T_{\lambda}\mathbf{a}^{*}_{m}\big{\rangle}_{\ell^{2}_{{}_{N}}(\Lambda)}\,T_{\lambda}\mathbf{b}_{m}\quad\text{for
each $\mathbf{c}\in\ell^{2}_{{}_{N}}(\Lambda)$}\,.$

Observe that a possible left-inverse $\widehat{B}(\xi)$ of the matrix
$\widehat{A}(\xi)$ is given by its Moore-Penrose pseudo-inverse
$\widehat{A}(\xi)^{\dagger}=\big{[}\widehat{A}(\xi)^{*}\widehat{A}(\xi)\big{]}^{-1}\widehat{A}(\xi)^{*}$,
a.e. $\xi\in\widehat{\Lambda}$.

###  3.4 The sampling results

Next we prove the main sampling result in this paper:

######  Theorem 4.

Suppose that for each $T\in V_{\bf S}^{2}$ we consider the samples defined by
(6), and such that the matrix $A=[a_{m,n}]$, where
$a_{m,n}(\lambda)=\big{\langle}\alpha_{-\lambda}(S_{n})g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}$,
$\lambda\in\Lambda$, satisfies conditions in Eq. (9). Then, there exist $M\geq
N$ elements $H_{m}\in V_{\bf S}^{2}$, $m=1,2,\dots,M$, such that the sampling
formula

$T=\sum_{m=1}^{M}\sum_{\lambda\in\Lambda}s_{{}_{T,m}}(\lambda)\,\alpha_{\lambda}(H_{m})\quad\text{in
$\mathcal{H}\mathcal{S}$-norm}$ (10)

holds for each $T\in V_{\bf S}^{2}$ where
$\\{\alpha_{\lambda}(H_{m})\\}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$ is a frame
for $V_{\bf S}^{2}$. The convergence of the series is unconditional in
Hilbert-Schmidt norm.

Moreover, the $\ell^{2}$-norm of the samples $\|{\bf
s}_{{}_{T}}\|_{\ell^{2}_{{}_{M}}}$ defines an equivalent norm to
$\|T\|_{{}_{\mathcal{H}\mathcal{S}}}$ in $V_{\bf S}^{2}$, and for each $f\in
L^{2}(\mathbb{R}^{d})$ we have the pointwise expansion

$Tf=\sum_{m=1}^{M}\sum_{\lambda\in\Lambda}s_{{}_{T,m}}(\lambda)\,\alpha_{\lambda}(H_{m})f\quad\text{in
$L^{2}(\mathbb{R}^{d})$}\,.$

###### Proof.

Under the hypotheses of the theorem the sequence
$\big{\\{}T_{\lambda}\,\mathbf{a}^{*}_{m}\big{\\}}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$
is a frame for $\ell^{2}_{{}_{N}}(\Lambda)$, and we can consider a dual frame
$\big{\\{}T_{\lambda}\,\mathbf{b}_{m}\big{\\}}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$
with the same structure. As a consequence, for each
$T=\sum_{n=1}^{N}\sum_{\lambda\in\Lambda}c_{n}(\lambda)\,\alpha_{\lambda}(S_{n})$
in $V_{\bf S}^{2}$ we have

$\mathbf{c}=\sum_{m=1}^{M}\sum_{\lambda\in\Lambda}\big{\langle}\mathbf{c},T_{\lambda}\mathbf{a}^{*}_{m}\big{\rangle}_{\ell^{2}_{{}_{N}}(\Lambda)}\,T_{\lambda}\mathbf{b}_{m}=\sum_{m=1}^{M}\sum_{\lambda\in\Lambda}s_{{}_{T,m}}(\lambda)\,T_{\lambda}\mathbf{b}_{m}\quad\text{in
$\ell^{2}_{{}_{N}}(\Lambda)$}\,,$ (11)

where
$\mathbf{c}=(c_{1},c_{2},\dots,c_{N})^{\top}\in\ell^{2}_{{}_{N}}(\Lambda)$.
Notice that the fact that
$\big{\\{}T_{\lambda}\,\mathbf{a}^{*}_{m}\big{\\}}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$
is a frame for $\ell^{2}_{{}_{N}}(\Lambda)$ and the isomorphism
$\mathcal{T}_{\bf S}$ in Eq. (5) give the equivalence of the norms.

The isomorphism $\mathcal{T}_{\bf S}$ defined by Eq. (5) applied in Eq. (11)
gives the sampling expansion (10), where $H_{m}=K_{h_{m}}\in V_{\bf S}^{2}$
with Kohn-Nirenberg symbol $h_{m}=\mathcal{T}_{\sigma_{\bf
S}}(\mathbf{b}_{m})\in V_{\sigma_{\bf S}}^{2}$, $m=1,2,\dots,M$. Furthermore,
since $\\{\alpha_{\lambda}(H_{m})\\}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$ is a
frame for $V_{\bf S}^{2}$ the convergence of the series in the Hilbert-Schmidt
norm is unconditional. Notice that
$\mathcal{T}_{\sigma_{S}}(T_{\lambda}\mathbf{b}_{m})=T_{\lambda}(\mathcal{T}_{\sigma_{S}}\mathbf{b}_{m})=T_{\lambda}(h_{m})$,
where the same symbol $T_{\lambda}$ denotes both the translation by $\lambda$
in $\ell^{2}_{{}_{N}}(\Lambda)$ and in $L^{2}(\mathbb{R}^{2d})$ respectively.
Notice that if
$\mathbf{b}_{m}=\big{(}b_{1,m}(\lambda),b_{2,m}(\lambda),\dots,b_{N,m}(\lambda)\big{)}^{\top}$,
then

$H_{m}=\sum_{n=1}^{N}\sum_{\lambda\in\Lambda}b_{n,m}(\lambda)\,\alpha_{\lambda}(S_{n})\,,\quad
m=1,2,\dots,M\,.$

Since convergence in $\mathcal{H}\mathcal{S}$-norm implies convergence in
operator norm we deduce the pointwise expansion for each $f\in
L^{2}(\mathbb{R}^{d})$. ∎

Observe that, due to conditions (9) in Theorem 4 we have necessarily $M\geq
N$. Whenever $M>N$, there are infinite dual frames
$\big{\\{}T_{\lambda}\,\mathbf{b}_{m}\big{\\}}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$
of
$\big{\\{}T_{\lambda}\,\mathbf{a}^{*}_{m}\big{\\}}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$
given by the samples (7). They are obtained from the left-inverses
$\widehat{B}(\xi)$ of $\widehat{A}(\xi)$ which are deduced, from the Moore-
Penrose pseudo-inverse $\widehat{A}(\xi)^{\dagger}$, as the $N\times M$
matrices

$\widehat{B}(\xi):=\widehat{A}(\xi)^{\dagger}+C(\xi)\big{[}\mathbb{I}_{{}_{M}}-\widehat{A}(\xi)\widehat{A}(\xi)^{\dagger}\big{]}\,,\text{\,
\,a.e. $\xi\in\widehat{\Lambda}$}\,,$

where $C$ denotes any $N\times M$ matrix with entries in
$L^{\infty}(\widehat{\Lambda})$.

More can be said in case $M=N$:

######  Corollary 5.

In case $M=N$, assume that the conditions

$0<\operatorname*{ess\,inf}_{\xi\in\widehat{\Lambda}}\big{|}\det[\widehat{A}(\xi)]\big{|}\leq\operatorname*{ess\,sup}_{\xi\in\widehat{\Lambda}}\big{|}\det[\widehat{A}(\xi)]\big{|}<+\infty$
(12)

hold. Then, there exist $N$ unique elements $H_{n}$, $n=1,2,\dots,N$, in
$V_{\bf S}^{2}$ such that the associated sequence
$\big{\\{}\alpha_{\lambda}(H_{n})\big{\\}}_{\lambda\in\Lambda;\,n=1,2,\dots,N}$
is a Riesz basis for $V_{\bf S}^{2}$ and the sampling formula

$T=\sum_{n=1}^{N}\sum_{\lambda\in\Lambda}s_{{}_{T,n}}(\lambda)\,\alpha_{\lambda}(H_{n})\quad\text{in
$\mathcal{H}\mathcal{S}$-norm}$

holds for each $T\in V_{\bf S}^{2}$. Moreover, the interpolation property
$\big{\langle}\alpha_{-\lambda}(H_{m})g_{n},\widetilde{g}_{n}\big{\rangle}=\delta_{m,n}\delta_{\lambda,0}$,
where $\lambda\in\Lambda$ and $m,n=1,2,\dots,N$, holds.

###### Proof.

In this case, the square matrix $\widehat{A}(\xi)$ is invertible and the
statement (12) in corollary is equivalent to condition
$0<\alpha_{A}\leq\beta_{A}<+\infty$ in (9); besides, any Riesz basis has a
unique dual basis. The uniqueness of the coefficients in a Riesz basis
expansion gives the interpolation property. ∎

In particular, for the case $N=M=1$ we have:

######  Corollary 6.

Assume that the sequence $\mathbf{a}=\\{a(\lambda)\\}_{\lambda\in\Lambda}$,
where
$a(\lambda)=\big{\langle}\alpha_{-\lambda}(S)g,\widetilde{g}\,\big{\rangle}_{L^{2}(\mathbb{R}^{d})}$,
$\lambda\in\Lambda$, for a fixed pair of functions $g,\widetilde{g}\in
L^{2}(\mathbb{R}^{d})$, satisfies the conditions

$0<\operatorname*{ess\,inf}_{\xi\in\widehat{\Lambda}}|\mathcal{F}_{s}^{\Lambda}(\mathbf{a})(\xi)|\leq\operatorname*{ess\,sup}_{\xi\in\widehat{\Lambda}}|\mathcal{F}_{s}^{\Lambda}(\mathbf{a})(\xi)|<\infty\,.$
(13)

Then, there exists a unique $H\in V^{2}_{{}_{S}}$ such that the sequence
$\big{\\{}\alpha_{\lambda}(H)\big{\\}}_{\lambda\in\Lambda}$ is a Riesz basis
for $V_{{}_{S}}^{2}$ and the sampling formula

$T=\sum_{\lambda\in\Lambda}\big{\langle}\alpha_{-\lambda}(T)g,\widetilde{g}\,\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\,\alpha_{\lambda}(H)\quad\text{in
$\mathcal{H}\mathcal{S}$-norm}$

holds for each $T\in V_{{}_{S}}^{2}$. Moreover, the interpolation property
$\big{\langle}\alpha_{-\lambda}(H)g,\widetilde{g}\,\big{\rangle}=\delta_{\lambda,0}$,
$\lambda\in\Lambda$, holds; in particular,
$\big{\langle}Hg,\widetilde{g}\,\big{\rangle}=1$.

It is worth to remark that in the above sampling result is not necessary that
the operators in $V_{{}_{S}}^{2}$ have a bandlimited Kohn-Nirenberg symbol as
in Ref. [18, Theorem 2].

The bandlimited case is obtained as a particular case. Let
$\Lambda=a\mathbb{Z}^{d}\times b\mathbb{Z}^{d}$ be a lattice in
$\mathbb{R}^{2d}$ with $a,b>0$. Assume that the generator $S$ of
$V^{2}_{{}_{S}}$ is a bandlimited operator to
$Q:=[\frac{-1}{2a},\frac{1}{2a}]^{d}\times[\frac{-1}{2b},\frac{1}{2b}]^{d}$,
i.e., it belongs to
$OPW^{2}(Q):=\\{T\in\mathcal{H}\mathcal{S}(\mathbb{R}^{d})\,:\,\text{supp\,}\widehat{\sigma}_{{}_{T}}\subseteq
Q\\}$. Then any $T\in V^{2}_{{}_{S}}$ also belongs to $OPW^{2}(Q)$. In case
conditions (13) are satisfied, any $T\in V^{2}_{{}_{S}}$ can be recovered from
its diagonal channel samples as

$T=\sum_{\lambda\in\Lambda}\big{\langle}T\pi(\lambda)g,\pi(\lambda)\widetilde{g}\,\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\,\alpha_{\lambda}(H)\quad\text{in
$\mathcal{H}\mathcal{S}$-norm}\,,$

where $H=\sum_{\lambda\in\Lambda}b(\lambda)\,\alpha_{\lambda}(S)$ in
$V^{2}_{{}_{S}}$ is obtained from the sequence
$\mathbf{b}=\\{b(\lambda)\\}_{\lambda\in\Lambda}$ in $\ell^{2}(\Lambda)$ such
that
$\mathcal{F}_{s}^{\Lambda}(\mathbf{b})(\xi)\,\mathcal{F}_{s}^{\Lambda}(\mathbf{a})(\xi)=1$,
a.e. $\xi\in\widehat{\Lambda}$.

In Ref. [18] the reconstruction of pseudodifferential operators with a
bandlimited Kohn-Nirenberg symbol is considered. In particular, Theorem 2 of
the same reference proves that, under some appropriate assumptions, for any
$T\in OPW^{2}(Q)$ we have

$\sigma_{{}_{T}}=\frac{1}{(ab)^{d}}\sum_{\lambda\in\Lambda}\big{\langle}T\pi(\lambda)g,\pi(\lambda)g\,\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\,T_{\lambda}\big{(}{\rm
sinc}_{a,b}\ast k\big{)}\quad\text{in $L^{2}(\mathbb{R}^{2d})$}\,,$

where the function $k$, independent of $T$, belongs to
$L^{1}(\mathbb{R}^{2d})$ and $\rm{sinc}_{a,b}$ denotes the sinc function
adapted to the lattice $\Lambda=a\mathbb{Z}^{d}\times b\mathbb{Z}^{d}$, namely

${\rm sinc}_{a,b}(x)=\prod_{j=1}^{d}\dfrac{\sin\pi ax_{j}}{\pi
ax_{j}}\prod_{j=d+1}^{2d}\dfrac{\sin\pi bx_{j}}{\pi bx_{j}}\,,\quad
x\in\mathbb{R}^{2d}\,.$

Using the Kohn-Nirenberg transfom, the above sampling formula for
$\sigma_{{}_{T}}$ can be written as

$T=\frac{1}{(ab)^{d}}\sum_{\lambda\in\Lambda}\big{\langle}T\pi(\lambda)g,\pi(\lambda)g\,\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\,\alpha_{\lambda}(k\ast
K_{\rm{sinc}_{a,b}})\quad\text{in $\mathcal{H}\mathcal{S}$-norm}\,,$

where $k\ast K_{{\rm sinc}_{a,b}}$ denotes the Hilbert-Schmidt operator
obtained from the convolution of the function $k$ and the operator $K_{{\rm
sinc}_{a,b}}$; we have also used the following result:

######  Lemma 7.

Let $K_{f}$ be an operator in $\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ with
Kohn-Nirenberg symbol $f\in L^{2}(\mathbb{R}^{2d})$, and let $g$ a function in
$L^{1}(\mathbb{R}^{2d})$. Then we have that $K_{g\ast f}=g\ast K_{f}$.

###### Proof.

Recall that the convolution $g\ast K_{f}$ is the operator in
$\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$ defined by the operator-valued
integral (in weak sense)

$g\ast K_{f}=\int_{\mathbb{R}^{2d}}g(z)\,\alpha_{z}(K_{f})dz\,,$

i.e.,

$\Big{\langle}\big{(}\int_{\mathbb{R}^{2d}}g(z)\,\alpha_{z}(K_{f})dz\big{)}\varphi,\psi\Big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\int_{\mathbb{R}^{2d}}g(z)\,\langle\alpha_{z}(K_{f})\varphi,\psi\rangle
dz\,,\quad\varphi,\psi\in L^{2}(\mathbb{R}^{d})\,.$

See the details in Refs. [21, 24, 26]. Since the map
$\mathcal{K}:\mathcal{H}\mathcal{S}(\mathbb{R}^{d})\rightarrow
L^{2}(\mathbb{R}^{2d})$ such that $\mathcal{K}(K_{f})=f$ is a unitary operator
and bounded operators conmute with convergent integrals [21, Proposition 2.4]
we get

$\mathcal{K}(g\ast
K_{f})=\int_{\mathbb{R}^{2d}}g(z)\,\mathcal{K}\big{(}\alpha_{z}(K_{f})\big{)}dz=\int_{\mathbb{R}^{2d}}g(z)\,\mathcal{K}\big{(}K_{T_{z}f}\big{)}dz=\int_{\mathbb{R}^{2d}}g(z)\,f(\cdot-z)dz=g\ast
f\,,$

that is, $K_{g\ast f}=g\ast K_{f}$. ∎

In the same manner we can consider average sampling in $V_{\bf S}^{2}$.
Namely, for any $T\in V_{\bf S}^{2}$, its average samples at $\Lambda$ are
defined by

$\big{\langle}T,\alpha_{\lambda}(Q_{m})\big{\rangle}_{{}_{\mathcal{H}\mathcal{S}}}\,,\quad\lambda\in\Lambda\,,\,\,\,m=1,2,\dots,M\,,$

from $M$ fixed operators $Q_{1},Q_{2},\dots,Q_{M}$ in
$\mathcal{H}\mathcal{S}(\mathbb{R}^{d})$, not necessarily in $V_{\bf S}^{2}$.
Observe that, having in mind Eq. (8) in Lemma 3, the diagonal channel samples
defined in Eq. (6) are a particular case of average sampling where
$Q_{m}=\widetilde{g}_{m}\otimes g_{m}$, $m=1,2,\dots,M$. The average samples
of any $T=\sum_{n=1}^{N}\sum_{\mu\in\Lambda}c_{n}(\mu)\,\alpha_{\mu}(S_{n})$
can be also expressed as a discrete convolution system in
$\ell^{2}_{{}_{N}}(\Lambda)$. Indeed, for $m=1,2,\dots,M$ we have

$\begin{split}\big{\langle}T,\alpha_{\lambda}(Q_{m})\big{\rangle}_{{}_{\mathcal{H}\mathcal{S}}}=&\big{\langle}\sigma_{{}_{T}},T_{\lambda}\sigma_{{}_{Q_{m}}}\big{\rangle}_{{}_{L^{2}(\mathbb{R}^{2d})}}=\big{\langle}\sum_{n=1}^{N}\sum_{\mu\in\Lambda}c_{n}(\mu)\,T_{\mu}\sigma_{{}_{S_{n}}},T_{\lambda}\sigma_{{}_{Q_{m}}}\big{\rangle}_{{}_{L^{2}(\mathbb{R}^{2d})}}\\\
=&\sum_{n=1}^{N}\sum_{\mu\in\Lambda}c_{n}(\mu)\langle
T_{\mu}\sigma_{{}_{S_{n}}},T_{\lambda}\sigma_{{}_{Q_{m}}}\rangle_{{}_{L^{2}(\mathbb{R}^{2d})}}=\sum_{n=1}^{N}\sum_{\mu\in\Lambda}c_{n}(\mu)\langle\sigma_{{}_{S_{n}}},T_{\lambda-\mu}\sigma_{{}_{Q_{m}}}\rangle_{{}_{L^{2}(\mathbb{R}^{2d})}}\\\
=&\sum_{n=1}^{N}\big{(}a_{m,n}\ast_{\Lambda}c_{n}\big{)}(\lambda)\,,\quad\lambda\in\Lambda\,,\end{split}$

where
$a_{m,n}(\mu):=\big{\langle}\sigma_{{}_{S_{n}}},T_{\mu}\sigma_{{}_{Q_{m}}}\big{\rangle}_{{}_{L^{2}(\mathbb{R}^{2d})}}=\big{\langle}S_{n},\alpha_{\mu}(Q_{m})\big{\rangle}_{{}_{\mathcal{H}\mathcal{S}}}$,
$\mu\in\Lambda$, and $\sigma_{{}_{S_{n}}}$, $\sigma_{{}_{Q_{m}}}$ are the
Kohn-Nirenberg symbols of $S_{n}$, $Q_{m}$ respectively.

Observe that, for each $m=1,2,\dots,M$ and $n=1,2,\dots,N$, the sequence
$\\{a_{m,n}(\lambda)\\}_{\lambda\in\Lambda}$ belongs to $\ell^{2}(\Lambda)$
since, in particular,
$\\{T_{\lambda}\sigma_{{}_{S_{n}}}\\}_{\lambda\in\Lambda;\,n=1,2,\dots,N}$ is
a Bessel sequence in $L^{2}(\mathbb{R}^{2d})$.

######  Corollary 8.

Assume that the matrix $A=[a_{m,n}]$ with entries
$a_{m,n}(\lambda)=\big{\langle}S_{n},\alpha_{\lambda}(Q_{m})\big{\rangle}_{{}_{\mathcal{H}\mathcal{S}}}$,
$\lambda\in\Lambda$, satisfies conditions in (9). Then, there exist $M\geq N$
operators $H_{m}\in V_{\bf S}^{2}$, $m=1,2,\dots,M$, such that the sampling
formula

$T=\sum_{m=1}^{M}\sum_{\lambda\in\Lambda}\big{\langle}T,\alpha_{\lambda}(Q_{m})\big{\rangle}_{{}_{\mathcal{H}\mathcal{S}}}\,\alpha_{\lambda}(H_{m})\quad\text{in
$\mathcal{H}\mathcal{S}$-norm}$

holds for each $T\in V_{\bf S}^{2}$ where
$\\{\alpha_{\lambda}(H_{m})\\}_{\lambda\in\Lambda;\,m=1,2,\dots,M}$ is a frame
for $V_{\bf S}^{2}$. The convergence of the series is unconditional in
Hilbert-Schmidt norm.

The above sampling formula was obtained in Ref. [12] by using the Weyl symbols
of $S_{n}$ and $Q_{m}$ instead of their Kohn-Nirenberg symbols. Finally, it is
worth to mention that each sampling result in this section admit a kind of
converse result; see the details in Theorems 1-2 and Corollary 3 of Ref. [12].

#### An illustrative example

Assume that in $V_{\bf S}^{2}$ we have $N$ stable generators of the form
$S_{n}=\varphi_{n}\otimes\widetilde{\varphi}_{n}$ with
$\varphi_{n},\widetilde{\varphi}_{n}\in\mathcal{S}_{0}(\mathbb{R}^{d})$,
$n=1,2,\dots,N$. In this regard, note that in order to apply Theorem 2 we have
that $\mathcal{F}_{W}(\varphi_{n}\otimes\widetilde{\varphi}_{n})(z)={\rm
e}^{\pi ix\cdot\omega}\,V_{\widetilde{\varphi}_{n}}\varphi_{n}(z)$,
$z=(x,\omega)\in\mathbb{R}^{2d}$ (see Ref. [24]).

Next, for each $T\in V_{\bf S}^{2}$ we consider the diagonal channel samples
$\big{\langle}T\pi(\lambda)g_{m},\pi(\lambda)\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}$,
$\lambda\in\Lambda$ and $m=1,2,\dots M$, with
$g_{m},\widetilde{g}_{m}\in\mathcal{S}_{0}(\mathbb{R}^{d})$. In this case, for
$m=1,2,\dots,M$ and $n=1,2,\dots,N$, we get

$\begin{split}a_{m,n}(\lambda)&=\big{\langle}\alpha_{-\lambda}(\varphi_{n}\otimes\widetilde{\varphi}_{n})g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\big{\langle}\big{(}\varphi_{n}\otimes\widetilde{\varphi}_{n}\big{)}\pi(\lambda)g_{m},\pi(\lambda)\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\\\
&=\big{\langle}\langle\pi(\lambda)g_{m},\widetilde{\varphi}_{n}\rangle\varphi_{n},\pi(\lambda)\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\overline{V_{g_{m}}\widetilde{\varphi}_{n}(\lambda)}\,V_{\widetilde{g}_{m}}\varphi_{n}(\lambda)\,,\quad\lambda\in\Lambda\,.\end{split}$

From Proposition 4.1 in Ref. [24] we deduce that the sequences
$\big{\\{}a_{m,n}(\lambda)\big{\\}}_{\lambda\in\Lambda}$ belong to
$\ell^{1}(\Lambda)$ and, as a consequence, the entries in the transfer matrix
$\widehat{A}$ are continuous functions on the compact $\widehat{\Lambda}$. In
order to apply Theorem 4 conditions in Eq. (9) reduce to

$\det[\widehat{A}(\xi)^{*}\widehat{A}(\xi)]\neq 0\quad\text{for all
$\xi\in\widehat{\Lambda}$}\,.$

Under the above circumstances, any $T\in V_{\bf S}^{2}$, which is nothing but
$T=\sum_{n=1}^{N}\mathcal{G}_{\mathbf{c}_{n}}^{\widetilde{\varphi}_{n},\varphi_{n}}$
a finite sum of Gabor multipliers, can be recovered, in a stable way, from its
diagonal channel samples
$\big{\langle}T\pi(\lambda)g_{m},\pi(\lambda)\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}$,
$\lambda\in\Lambda$ and $m=1,2,\dots,M$.

###  3.5 Sampling in a sub-lattice of $\Lambda$

Let $\Lambda^{\prime}$ be a sub-lattice of $\Lambda$ with finite index $L$,
i.e., the quotient group $\Lambda/\Lambda^{\prime}$ has finite order $L$. We
consider $\\{\lambda_{1},\lambda_{2},\dots,\lambda_{L}\\}$ a set of
representatives of the cosets of $\Lambda^{\prime}$. That is, the lattice
$\Lambda$ be decomposed as

$\Lambda=\bigcup_{l=1}^{L}(\lambda_{l}+\Lambda^{\prime})\quad\text{with}\quad(\lambda_{l}+\Lambda^{\prime})\cap(\lambda_{l^{\prime}}+\Lambda^{\prime})=\varnothing\,\,\text{for
$l\neq l^{\prime}$}\,.$

Thus, the space $V_{\bf S}^{2}$ can be written as

$\begin{split}V_{\bf
S}^{2}&=\Big{\\{}\sum_{n=1}^{N}\sum_{\lambda\in\Lambda}c_{n}(\lambda)\,\alpha_{\lambda}(S_{n})\,:\,c_{n}\in\ell^{2}(\Lambda)\Big{\\}}=\Big{\\{}\sum_{n=1}^{N}\sum_{l=1}^{L}\sum_{\mu\in\Lambda^{\prime}}c_{n}(\lambda_{l}+\mu)\,\alpha_{\lambda_{l}+\mu}(S_{n})\Big{\\}}\\\
&=\Big{\\{}\sum_{n=1}^{N}\sum_{l=1}^{L}\sum_{\mu\in\Lambda^{\prime}}c_{nl}(\mu)\,\alpha_{\mu}(S_{nl})\,:\,c_{nl}\in\ell^{2}(\Lambda^{\prime})\Big{\\}}\,,\end{split}$

where $c_{nl}(\mu):=c_{n}(\lambda_{l}+\mu)$ and
$S_{nl}:=\alpha_{\lambda_{l}}(S_{n})$, and the new index $nl$ goes from $11$
to $NL$. As a consequence, the subspace $V_{\bf S}^{2}$ can be rewritten as
$V_{\widetilde{\bf S}}^{2}$ with $NL$ generators $\widetilde{\bf
S}=\\{S_{nl}\\}$ and coefficients $c_{nl}$ in $\ell^{2}(\Lambda^{\prime})$.

Let
$T=\sum_{n=1}^{N}\sum_{l=1}^{L}\sum_{\nu\in\Lambda^{\prime}}c_{nl}(\nu)\,\alpha_{\nu}(S_{nl})$
be in $V_{\bf S}^{2}$; its samples
$\big{\langle}\alpha_{-\mu}(T)g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}$,
$\mu\in\Lambda^{\prime}$, can be expressed by

$\begin{split}s_{{}_{T,m}}(\mu):=&\big{\langle}\alpha_{-\mu}(T)g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\Big{\langle}\sum_{n=1}^{N}\sum_{l=1}^{L}\sum_{\nu\in\Lambda^{\prime}}c_{nl}(\mu)\,\alpha_{\nu-\mu}(S_{nl})g_{m},\widetilde{g}_{m}\Big{\rangle}_{L^{2}(\mathbb{R}^{d})}\\\
=&\sum_{n=1}^{N}\sum_{l=1}^{L}\sum_{\nu\in\Lambda^{\prime}}c_{nl}(\nu)\big{\langle}\alpha_{\nu-\mu}(S_{nl})g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}=\sum_{n=1}^{N}\sum_{l=1}^{L}\big{(}a_{m,nl}\ast_{\Lambda^{\prime}}c_{nl}\big{)}(\mu)\,,\quad\mu\in\Lambda^{\prime}\,,\end{split}$

where
$a_{m,nl}(\nu):=\big{\langle}\alpha_{-\nu}(S_{nl})g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}$,
$\nu\in\Lambda^{\prime}$. Hence, Theorem 4 gives:

######  Corollary 9.

Let $A=[a_{m,nl}]$ be the $M\times NL$ matrix with entries

$a_{m,nl}(\nu)=\big{\langle}\alpha_{-\nu}(S_{nl})g_{m},\widetilde{g}_{m}\big{\rangle}_{L^{2}(\mathbb{R}^{d})}\,,\quad\nu\in\Lambda^{\prime}\,,$

for $m=1,2,\dots,M$ and $nl=11,12,\dots,NL$. Assume that $A$ satisfies
conditions in (9) with respect to the dual $\widehat{\Lambda^{\prime}}$. Then,
there exist $M\geq NL$ operators $H_{m}\in V_{\bf S}^{2}$, $m=1,2,\dots,M$,
such that the sampling formula

$T=\sum_{m=1}^{M}\sum_{\mu\in\Lambda^{\prime}}s_{{}_{T,m}}(\mu)\,\alpha_{\mu}(H_{m})\quad\text{in
$\mathcal{H}\mathcal{S}$-norm}$

holds for each $T\in V_{\bf S}^{2}$ where
$\\{\alpha_{\mu}(H_{m})\\}_{\mu\in\Lambda^{\prime};\,m=1,2,\dots,M}$ is a
frame for $V_{\bf S}^{2}$. The convergence of the series is unconditional in
Hilbert-Schmidt norm.

Acknowledgments: The author thanks Universidad Carlos III de Madrid for
granting him a sabbatical year in 2020-21. This work has been supported by the
grant MTM2017-84098-P from the Spanish Ministerio de Economía y Competitividad
(MINECO).

## References

  * [1] A. Aldroubi, Q. Sun and W. S. Tang.  Convolution, average sampling, and a Calderon resolution of the identity for shift-invariant spaces.  J. Fourier Anal. Appl., 11(2):215–244 (2005).
  * [2] J.J. Benedetto and G. E. Pfander.  Frame expansions for Gabor multipliers.  Appl. Comput. Harmon. Anal., 20(1):26–40 (2006).
  * [3] O. Christensen.  An Introduction to Frames and Riesz Bases, 2nd ed.,  Birkhäuser, Basel (2016).
  * [4] J. B. Conway.  A Course in Operator Theory,  AMS, Providence RI (2000).
  * [5] A. Deitmar and S. Echterhoff.  Principles of Harmonic Analysis, 2nd ed.,  Universitext, Springer (2014).
  * [6] H. G. Feichtinger.  Spline-type spaces in Gabor analysis.  In Wavelet Analysis, D-X. Zhou Ed., pp. 100–122. World Sci. Publ., River Edge, NJ (2002).
  * [7] H. G. Feichtinger, F. Luef and T. Wherter.  A guided tour from linear algebra to the foundations of Gabor analysis.  In Gabor and Wavelet frames, Say Song Goh et al. (Eds.), pp. 1–49, Lecture Notes Series, IMS, National University of Singapore: Vol. 10, World Scientific (2007).
  * [8] H. R. Fernández-Morales, A. G. García, M. A. Hernández-Medina and M. J. Muñoz-Bouzo.  Generalized sampling: from shift-invariant to $U$-invariant spaces.  Anal. Appl., 13(3):303–329 (2015).
  * [9] G. B. Folland.  Harmonic Analysis in Phase Space,  Princeton University Press, Princeton (1989).
  * [10] G. B. Folland.  A Course in Abstract Harmonic Analysis,  CRC Press (1995).
  * [11] H. Führ.  Abstract Harmonic Analysis of Continuous Wavelet Transform.  Springer (2005).
  * [12] A. G. García.  Average sampling in certain subspaces of Hilbert-Schmidt operators on $L^{2}(\mathbb{R}^{d})$.  ArXiv: 2011.05871v2 (2021).
  * [13] A. G. García and G. Pérez-Villalón.  Dual frames in ${L}^{2}(0,1)$ connected with generalized sampling in shift-invariant spaces.  Appl. Comput. Harmon. Anal., 20(3):422–433 (2006).
  * [14] A. G. García, M. A. Hernández-Medina and G. Pérez-Villalón.  Generalized sampling in shift-invariant spaces with multiple stable generators.  J. Math. Anal. Appl., 337:69–84 (2008).
  * [15] A. G. García, M.A. Hernández-Medina and G. Pérez-Villalón.  Convolution systems on discrete abelian groups as a unifying strategy in sampling theory.  Results Math., 75:40 (2020).
  * [16] K. Gröchenig.  Foundations of Time-Frequency Analysis.  Birkhäuser, Basel (2001).
  * [17] K. Gröchenig and C. Heil.  Modulation spaces and pseudodifferential operators.  Integr. Equ. Oper. Theory, 34(4):439–457 (1999).
  * [18] K. Gröchenig and E. Pauwels.  Uniqueness and reconstruction theorems for pseudodifferential operators with a bandlimited Kohn-Nirenberg symbol.  Adv. Comput. Math., 40:49–63 (2014).
  * [19] M. S. Jakobsen.  On a (no longer) new Segal Algebra: a review of the Feichtinger Algebra.  J. Fourier Anal. Appl., 24:1579–1660 (2018).
  * [20] F. Krahmer and G. E. Pfander.  Local sampling and approximation of operators.  Construct. Approx., 39(3):541–572 (2014).
  * [21] F. Luef and E. Skrettingland.  Convolutions for localizations operators.  J. Math. Pures Appl., 118:288–316 (2018).
  * [22] G. E. Pfander.  Sampling of operators.  J. Fourier Anal. Appl., 19:612–650 (2013).
  * [23] G. E. Pfander and D. F. Walnut.  Sampling and reconstruction of operators.  IEEE Trans, Inform. Theory, 62(1):435–458 (2016).
  * [24] E. Skrettingland.  Quantum harmonic analysis on lattices and Gabor multipliers.  J. Fourier Anal. Appl., 26:48 (2020).
  * [25] T. Strohmer.  Pseudodifferential operators and Banach algebras in mobile communications.  Appl. Comput. Harmon. Anal., 20:237–249 (2006).
  * [26] R. F. Werner.  Quantum harmonic analysis on phase space.  J. Math. Phys., 25(5):1404–1411 (1984).


Condensed interior-point methods: porting reduced-space approaches on GPU hardware

Pacaud et al.

François Pacaud Sungho Shin Michel Schanen Daniel Adrian Maldonado Mihai Anitescu
Mihai Anitescu dedicates this work to the 70-th birthday of Florian Potra. Florian, thank you for the great contributions to optimization in general, and interior point methods in particular, and for initiating me and many others in them.

All the authors are affiliated to Argonne National Laboratory.

The interior-point method (IPM) has become the workhorse
method for nonlinear programming. The performance of IPM is
directly related to the linear solver employed to factorize the Karush–Kuhn–Tucker (KKT) system
at each iteration of the algorithm.
When solving large-scale nonlinear problems, state-of-the art IPM solvers
rely on efficient sparse linear solvers to solve the KKT system.

Instead, we propose a novel reduced-space
IPM algorithm that condenses the KKT system into a dense matrix
whose size is proportional to the number of degrees of freedom in the problem.
Depending on where the reduction occurs we derive two variants of the reduced-space
method: linearize-then-reduce and reduce-then-linearize. We adapt their
workflow so that the vast majority of computations are accelerated on GPUs.

We provide extensive numerical results
on the optimal power flow problem, comparing our GPU-accelerated reduced
space IPM with Knitro and a hybrid full space IPM algorithm.
By evaluating the derivatives on the GPU and solving the KKT system
on the CPU, the hybrid solution is already significantly faster than the CPU-only solutions.
The two reduced-space algorithms go one step further by solving the KKT system entirely
on the GPU.
As expected, the performance of the two reduction algorithms depends
intrinsically on the number of available degrees of freedom:
their performance is poor when the problem has many degrees of freedom,
but the two algorithms are up to 3 times faster than Knitro as soon as the
relative number of degrees of freedom becomes smaller.

§ CONCLUSION

This paper has presented an efficient implementation of the IPM
on GPU architectures, based on a Schur reduction of the underlying KKT system.
We have derived two practical algorithms, linearize-then-reduce and
reduce-then-linearize, adapted their workflows to be efficient when
the vast majority of their computation is run on the GPU, and detailed their respective performance
on different large-scale OPF instances.

We also discussed the benefits of a hybrid full space IPM solver — computing the
derivatives on the GPU and the linear algebra on the CPU — and demonstrated
that this approach generally outperforms Knitro, running exclusively on the CPU.

The relative performance of the reduced-space algorithms is highly dependent on
the ratio of controls with respect to the total number of variables.
Their performance lags behind both
Knitro and the hybrid full space solver when the problem has many
control variables (as it is the case on the MATPOWER benchmark)
but is significantly ahead – up to a factor of 3 – when
the problem has a relatively lower number of control variables.

Moreover, the reduce-then-linearize algorithm has the added
benefit of producing a feasible solution to the power flow equations at
any iteration, which makes it a great candidate for real time applications.
To improve the performance of reduction algorithms, we believe the most
important item is to alleviate the dependence on the
number of control variables. We plan to explore a way to accelerate the
reduction by exploiting the exponentially decaying structure of the reduced
Hessian [34].

§ ACKNOWLEDGMENTS

We thank
Juraj Kardoš for reviving the interest in reduced-space methods and
Kasia Świrydowicz for pointing us to the cusolverRF library.
This research was supported by the Exascale Computing Project (17-SC-20-SC),
a joint project of the U.S. Department of Energy’s Office of Science and
National Nuclear Security Administration, responsible for delivering a
capable exascale ecosystem, including software, applications, and hardware
technology, to support the nation’s exascale computing imperative.
The material is based upon work supported in part by the U.S. Department of Energy, Office of Science, under contract  DE-AC02-06CH11357.

[1]
J. Abadie and J. Carpentier, Generalization of the Wolfe reduced
gradient method to the case of nonlinear constraints, 1969.

[2]
S. Babaeinejadsarookolaee, A. Birchfield, R. D. Christie, C. Coffrin,
C. DeMarco, R. Diao, M. Ferris, S. Fliscounakis, S. Greene, R. Huang,
et al., The power grid library for benchmarking AC optimal power flow
algorithms, arXiv preprint arXiv:1908.02788,  (2019).

[3]
L. T. Biegler, J. Nocedal, and C. Schmid, A reduced Hessian method
for large-scale constrained optimization, SIAM Journal on Optimization, 5
(1995), pp. 314–347.

[4]
G. Biros and O. Ghattas, Parallel
Lagrange–Newton–Krylov–Schur methods for PDE-constrained
optimization: Part I – The Krylov–Schur Solver, SIAM Journal on
Scientific Computing, 27 (2005), pp. 687–713.

[5]
R. Burchett, H. Happ, and D. Vierath, Quadratically convergent
optimal power flow, IEEE Transactions on Power Apparatus and Systems,
(1984), pp. 3267–3275.

[6]
M. B. Cain, R. P. O’neill, A. Castillo, et al., History of optimal
power flow and formulations, Federal Energy Regulatory Commission, 1 (2012),
pp. 1–36.

[7]
Y. Cao, A. Seth, and C. D. Laird, An augmented Lagrangian
interior-point approach for large-scale NLP problems on graphics processing
units, Computers & Chemical Engineering, 85 (2016), pp. 76–83.

[8]
A. M. Cervantes, A. Wächter, R. H. Tütüncü, and L. T.
Biegler, A reduced space interior point strategy for optimization of
differential algebraic systems, Computers & Chemical Engineering, 24
(2000), pp. 39–51.

[9]
T. F. Coleman and A. R. Conn, Nonlinear programming via an exact
penalty function: Asymptotic analysis, Mathematical Programming, 24 (1982),
pp. 123–136.

[10]
H. Dommel and W. Tinney, Optimal power flow solutions, IEEE
Transactions on Power Apparatus and Systems, PAS-87 (1968), pp. 1866–1876.

[11]
I. S. Duff, MA57—a code for the solution of sparse symmetric
definite and indefinite systems, ACM Transactions on Mathematical Software
(TOMS), 30 (2004), pp. 118–144.

[12]
I. Dunning, J. Huchette, and M. Lubin, JuMP: A modeling language
for mathematical optimization, SIAM review, 59 (2017), pp. 295–320.

[13]
R. Fletcher, Practical methods of optimization, John Wiley & Sons,

[14]
R. Fourer, D. M. Gay, and B. W. Kernighan, A modeling language for
mathematical programming, Management Science, 36 (1990), pp. 519–554.

[15]
S. Frank, I. Steponavice, and S. Rebennack, Optimal power flow: a
bibliographic survey I, Energy systems, 3 (2012), pp. 221–258.

[16]
D. Gabay, Minimizing a differentiable function over a differential
manifold, Journal of Optimization Theory and Applications, 37 (1982),
pp. 177–219.

[17]
A. Griewank and A. Walther, Evaluating derivatives: principles and
techniques of algorithmic differentiation, SIAM, 2008.

[18]
C. B. Gurwitz and M. L. Overton, Sequential quadratic programming
methods based on approximating a projected Hessian matrix, SIAM Journal on
Scientific and Statistical Computing, 10 (1989), pp. 631–653.

[19]
H. Hijazi, G. Wang, and C. Coffrin, Gravity: A mathematical modeling
language for optimization and machine learning, Machine Learning Open Source
Software Workshop at NeurIPS 2018,  (2018).
Available at <www.gravityopt.com>.

[20]
Q. Jiang and G. Geng, A reduced-space interior point method for
transient stability constrained optimal power flow, IEEE Transactions on
Power Systems, 25 (2010), pp. 1232–1240.

[21]
J. Kardos, D. Kourounis, and O. Schenk, Reduced-space interior point
methods in power grid problems, arXiv preprint arXiv:2001.10815,  (2020).

[22]
J. Kardos, D. Kourounis, O. Schenk, and R. Zimmerman, Complete
results for a numerical evaluation of interior point solvers for large-scale
optimal power flow problems, arXiv preprint arXiv:1807.03964,  (2018).

[23]
Y. Kim, F. Pacaud, K. Kim, and M. Anitescu, Leveraging GPU
batching for scalable nonlinear programming through massive Lagrangian
decomposition, arXiv preprint arXiv:2106.14995,  (2021).

[24]
D. Lee, K. Turitsyn, D. K. Molzahn, and L. A. Roald, Feasible path
identification in optimal power flow with sequential convex restriction,
IEEE Transactions on Power Systems, 35 (2020), pp. 3648–3659.

[25]
J. Nocedal and M. L. Overton, Projected Hessian updating
algorithms for nonlinearly constrained optimization, SIAM Journal on
Numerical Analysis, 22 (1985), pp. 821–850.

[26]
J. Nocedal and S. J. Wright, Numerical optimization, Springer
series in operations research, Springer, New York, 2nd ed ed., 2006.
OCLC: ocm68629100.

[27]
F. Pacaud, D. A. Maldonado, S. Shin, M. Schanen, and M. Anitescu, A
feasible reduced space method for real-time optimal power flow, arXiv
preprint arXiv:2110.02590,  (2021).

[28]
F. Pacaud, M. Schanen, D. A. Maldonado, A. Montoison, V. Churavy,
J. Samaroo, and M. Anitescu, Batched second-order adjoint sensitivity
for reduced space methods, arXiv preprint arXiv:2201.00241,  (2022).

[29]
J. Revels, M. Lubin, and T. Papamarkou, Forward-mode automatic
differentiation in Julia, arXiv preprint arXiv:1607.07892,  (2016).

[30]
J. B. Rosen, The gradient projection method for nonlinear
programming: Part II – Nonlinear constraints, Journal of the Society for
Industrial and Applied Mathematics, 9 (1961), pp. 514–532.

[31]
R. Sargent et al., Reduced-gradient and projection methods for
nonlinear programming, Numerical Methods for Constrained Optimization, 149

[32]
O. Schenk and K. Gärtner, Solving unsymmetric sparse systems of
linear equations with PARDISO, Future Generation Computer Systems, 20
(2004), pp. 475–487.

[33]
M. Schubiger, G. Banjac, and J. Lygeros, GPU acceleration of
ADMM for large-scale quadratic programming, Journal of Parallel and
Distributed Computing, 144 (2020), pp. 55–67.

[34]
S. Shin, M. Anitescu, and V. M. Zavala, Exponential decay of
sensitivity in graph-structured nonlinear programs, arXiv preprint
arXiv:2101.03067,  (2021).

[35]
S. Shin, C. Coffrin, K. Sundar, and V. M. Zavala, Graph-based
modeling and decomposition of energy infrastructures, IFAC-PapersOnLine, 54
(2021), pp. 693–698.

[36]
K. Świrydowicz, E. Darve, W. Jones, J. Maack, S. Regev, M. A.
Saunders, S. J. Thomas, and S. Peleš, Linear solvers for power
grid optimization problems: a review of GPU-accelerated linear solvers,
Parallel Computing,  (2021), p. 102870.

[37]
B. Tasseff, C. Coffrin, A. Wächter, and C. Laird, Exploring
benefits of linear solver parallelism on modern nonlinear optimization
applications, arXiv preprint arXiv:1909.08104,  (2019).

[38]
A. Wächter and L. T. Biegler, On the implementation of an
interior-point filter line-search algorithm for large-scale nonlinear
programming, Mathematical Programming, 106 (2006), pp. 25–57.

[39]
R. A. Waltz, J. L. Morales, J. Nocedal, and D. Orban, An interior
algorithm for nonlinear optimization that combines line search and trust
region steps, Mathematical Programming, 107 (2006), pp. 391–408.

[40]
R. D. Zimmerman, C. E. Murillo-Sánchez, and R. J. Thomas, MATPOWER: Steady-state operations, planning, and analysis tools for power
systems research and education, IEEE Transactions on Power Systems, 26
(2010), pp. 12–19.

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