Research Article
Modeling Sampling in Tensor Products of
Unitary Invariant Subspaces
Antonio G. García,
1
Alberto Ibort,
1
and María J. Muñoz-Bouzo
2
1
Departamento de Matem´ aticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, Legan´ es, 28911 Madrid, Spain
2
Departamento de Matem´ aticas Fundamentales, U.N.E.D., Senda del Rey 9, 28040 Madrid, Spain
Correspondence should be addressed to Antonio G. Garc´ ıa; agarcia@math.uc3m.es
Received 25 July 2016; Accepted 16 October 2016
Academic Editor: Hans G. Feichtinger
Copyright © 2016 Antonio G. Garc´ ıa et al. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Te use of unitary invariant subspaces of a Hilbert space H is nowadays a recognized fact in the treatment of sampling problems.
Indeed, shif-invariant subspaces of
2
(R) and also periodic extensions of fnite signals are remarkable examples where this occurs.
As a consequence, the availability of an abstract unitary sampling theory becomes a useful tool to handle these problems. In
this paper we derive a sampling theory for tensor products of unitary invariant subspaces. Tis allows merging the cases of
fnitely/infnitely generated unitary invariant subspaces formerly studied in the mathematical literature; it also allows introducing
the several variables case. As the involved samples are identifed as frame coefcients in suitable tensor product spaces, the relevant
mathematical technique is that of frame theory, involving both fnite/infnite dimensional cases.
1. Introduction
Sampling and reconstruction of functions in unitary invari-
ant subspaces of a separable Hilbert space bring a uni-
fed approach to sampling problems (see [1–5]). Indeed, it
englobes the most usual sampling settings such as sampling
in shif-invariant subspaces of
2
(R) (see, e.g., [6–17] and
references therein) or sampling periodic extensions of fnite
signals (see [3, 18]).
In a recent paper [19] it was shown how to extend sam-
pling reconstruction theorems to invariant subspaces of sep-
arable Hilbert spaces under a unitary representation of fnite
groups which are semidirect products with an Abelian factor.
Tis setting is appropriate for applications of the theory
beyond the domain of classical telecommunications to quan-
tum physics.
In this paper we go one step ahead by enlarging the class
of target spaces for sampling: we deal with tensor products
of diferent unitary invariant subspaces. Tis situation cor-
responds, for instance, to consider multichannel systems in
classical telecommunications or composite systems in the
case of quantum applications. Tus, in this setting, we are
able to gather problems of diverse nature by means of a simple
formalism involving tensor products and tensor operators in
Hilbert spaces. Namely, we frst consider an infnite -unitary
subspace
A
={∑
∈Z
:{
}∈ℓ
2
(Z)} (1)
in a Hilbert space H
1
and a fnite -unitary subspace
A
={
−1
∑
=0
:
∈ C} (2)
in a Hilbert space H
2
, to fnally obtain sampling formulas in
their tensor product A
,
fl A
⊗ A
.
Apart from tensor products in Hilbert spaces, the paper
involves the theory of frames. Concretely, in this situation, the
generalized samples will be expressed as frame coefcients
in an auxiliary Hilbert space
2
(0,1)⊗ℓ
2
(Z), where ℓ
2
(Z)
denotes the space of all -periodic complex sequences.
Continuing the line of inquiry of [2, 3], the problem reduces
Hindawi Publishing Corporation
Journal of Function Spaces
Volume 2016, Article ID 4573940, 14 pages
http://dx.doi.org/10.1155/2016/4573940