Journal of Function Spaces
Volume 2016 (2016), Article ID 4573940, 14 pages
http://dx.doi.org/10.1155/2016/4573940

Research Article

Modeling Sampling in Tensor Products of Unitary Invariant Subspaces

Antonio G. García,¹ Alberto Ibort,¹ and María J. Muñoz-Bouzo²

¹Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, Leganés, 28911 Madrid, Spain
²Departamento de Matemáticas Fundamentales, U.N.E.D., Senda del Rey 9, 28040 Madrid, Spain

Received 25 July 2016; Accepted 16 October 2016

Academic Editor: Hans G. Feichtinger

Copyright © 2016 Antonio G. García et al. This 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.

Abstract.

The use of unitary invariant subspaces of a Hilbert space is nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces of and also periodic extensions of finite 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. This allows merging the cases of finitely/infinitely generated unitary invariant subspaces formerly studied in the mathematical literature; it also allows introducing the several variables case. As the involved samples are identified as frame coefficients in suitable tensor product spaces, the relevant mathematical technique is that of frame theory, involving both finite/infinite dimensional cases.

1. Introduction

Sampling and reconstruction of functions in unitary invariant subspaces of a separable Hilbert space bring a unified approach to sampling problems (see [1–5]). Indeed, it englobes the most usual sampling settings such as sampling in shift-invariant subspaces of (see, e.g., [6–17] and references therein) or sampling periodic extensions of finite signals (see [3, 18]).

In a recent paper [19] it was shown how to extend sampling reconstruction theorems to invariant subspaces of separable Hilbert spaces under a unitary representation of finite groups which are semidirect products with an Abelian factor. This setting is appropriate for applications of the theory beyond the domain of classical telecommunications to quantum physics.

In this paper we go one step ahead by enlarging the class of target spaces for sampling: we deal with tensor products of different unitary invariant subspaces. This situation corresponds, for instance, to consider multichannel systems in classical telecommunications or composite systems in the case of quantum applications. Thus, 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 first consider an infinite -unitary subspace

in a Hilbert space and a finite -unitary subspace

in a Hilbert space , to finally obtain sampling formulas in their tensor product .

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 coefficients in an auxiliary Hilbert space , where denotes the space of all -periodic complex sequences. Continuing the line of inquiry of [2, 3], the problem reduces to find appropriate families of dual frames. By “appropriate” we mean that these dual frames have a nice structure taking care of the unitary invariance of the involved sampling subspaces as it will be discussed in the sequel.

Later on, the infinite-infinite and finite-finite generator cases will be considered too, that is, the situation where both invariant subspaces are generated by a sequence of vectors , , and and the simpler case where both subspaces are finite dimensional. Relevant examples of each situation will be discussed in detail.

The paper is organized as follows: for the sake of completeness we include in Section 2 the basics of frames and tensor products needed in the sequel. In Section 3 we focus on the above case that we call infinite-finite generators case; that is, we establish sampling formulas in the tensor product of two unitary invariant subspaces, one of them with an infinite generator and the other one finitely generated. First, we obtain appropriate expressions for the samples of any obtained from systems acting on ,

where , , , and . Here, , , denote fixed elements in and , , denote fixed elements in ; and are the sampling periods, where and is a divisor of and (see Section 3 for the details). Then we state the suitable isomorphism between and which allows transforming the derived frame expansions in into stable sampling formulas in having the form

where , , and , . We conclude the section giving a representative example arising from classical tomography (see, e.g., [20, 21]).

Sections 4 and 5 deal with the called infinite-infinite and finite-finite cases. They mimic the structure of Section 3 with auxiliary spaces and , respectively.

Finally, it is worth mentioning that for the sake of simplicity we only deal with the tensor product of two single generated unitary invariant subspaces; the same results apply for the tensor product of any finite number of multiple generated unitary invariant subspaces.

2. A Brief on Frames and Tensor Products

The frame concept was introduced by Duffin and Schaeffer in [22] while studying some problems in nonharmonic Fourier series; some years later it was revived by Daubechies et al. in [23]. Nowadays, frames have become a tool in pure and applied mathematics, computer science, physics, and engineering used to derive redundant, yet stable decompositions of a signal for analysis or transmission, while also promoting sparse expansions. Recall that a sequence is a frame for a separable Hilbert space if there exist two constants (frame bounds) such that

A sequence satisfying only the right-hand inequality is said to be a Bessel sequence for . Given a frame for the representation property of any vector as a series is retained, but, unlike the case of Riesz bases, the uniqueness of this representation (for overcomplete frames) is sacrificed. Suitable frame coefficients which depend continuously and linearly on are obtained by using dual frames of ; that is, is another frame for such that for each

Recall that a Riesz basis in a separable Hilbert space is the image of an orthonormal basis by means of a boundedly invertible operator; it is a particular case of frame: the so-called exact frame. Any Riesz basis has a unique biorthonormal (dual) Riesz basis ; that is, , such that expansion (6) holds for every . A Riesz sequence is a Riesz basis for its closed span. For more details on frames and Riesz bases theory see, for instance, the monograph [8] and references therein; see also [24] for finite frames.

Traditionally, frames were used in signal and image processing, nonharmonic analysis, data compression, and sampling theory, but nowadays frame theory plays also a fundamental role in a wide variety of problems in both pure and applied mathematics, computer science, physics, and engineering. The redundancy of frames, which gives flexibility and robustness, is the key to their significance for applications; see, for instance, the nice introduction in Chapter 1 of [24] and the references therein.

Next we briefly recall some basic facts about tensor products of Hilbert spaces which will be useful in the current work. Let and be two Hilbert spaces. Among the different ways of constructing tensor product spaces we adopt the model proposed in [25]. There, the tensor product is defined as the space of all antilinear maps such that for some orthonormal basis of of .

As for every the series and are independent of the orthonormal basis for ; then can be turned into an inner product space by defining the norm (and the associated inner product ). Indeed endowed with this inner product becomes a Hilbert space.

If and the tensor product is defined to be the rank one map such that for every .

Let , , and denote the spaces of all bounded linear operators on , , and , respectively. If and the tensor product is defined to be the bounded linear operator on such that for every .

For further information on tensor products of Hilbert spaces see [25–28]. In these sources the following results for tensor products needed in the sequel are to be found.

(i) and for any and .(ii) and for any , , , and .(iii)The linear span of is dense in .(iv)The tensor product of two orthonormal bases is an orthonormal basis.(v)The operator is invertible in if and only if each operator, and , is invertible in and , respectively.(vi)The tensor product of two sequences is a Riesz basis for if and only if each sequence is a Riesz basis for its corresponding Hilbert space.(vii)The tensor product of two Bessel sequences is a Bessel sequence for the corresponding Hilbert space.(viii)The tensor product of two sequences is a frame for if and only if each sequence is a frame for its corresponding Hilbert space.

Note that the fact that is the completion of the linear span of yields that any operator of , in particular the tensor product , is determined by its values on .

Finally, let and denote two -finite measure spaces, and then we have that via the identification .

3. Infinite-Finite Generators Case

Let and be two separable Hilbert spaces and and two unitary operators. Consider two elements and such that the sequence forms a Riesz sequence in (see [2, Theorem  2.1] for a necessary and sufficient condition), and there exists an such that , being the set linearly independent in (see [3, Proposition  1] for a necessary and sufficient condition). In the tensor product Hilbert space we consider the closed subspace

Since the sequence is a Riesz basis for the tensor product of the -invariant subspace of and the -invariant subspace of we deduce that , and it can be described as

We will refer to the vectors as the infinite-finite generators of the subspace in .

The Samples. Fix elements , where and . Consider two sampling periods and , where and is a divisor of ; denote . For each we introduce the sequence of its generalized samples

defined by

where , , , and . We will refer to any as a -system acting on the subspace .

The Isomorphism . Let denote the space of all -periodic sequences in with inner product and its canonical basis as . This space is isomorphic to the euclidean space ; along the paper we will identify sequences in with vectors in : any vector in defines the terms from to of the corresponding sequence in .

We introduce the isomorphism which maps the canonical orthonormal basis for the Hilbert space onto the Riesz basis for . That is,

where and .

It is clear that , where and denote the isomorphisms

The following shifting property will be crucial later in obtaining our sampling formulas.

Lemma 1. Let and . For and the shifting property

holds, where denotes the sequence obtained from as

Proof. Indeed, having in mind the shifting properties: for (see [2, Eq. ]) and for (see [3, Prop. ]), and the properties of the tensor product of operators, we have

An Expression for the Samples. A suitable expression for the samples given in (10) will allow us to obtain the reconstruction conditions of any in the subspace from the sequence of its samples (10). Indeed, for in we have

Now, consider , which is the element in such that . Thus, Plancherel identity for orthonormal bases gives

That is, for , , , and we have got the following expression for the samples:

where , the functions

belong to (recall that is a Riesz sequence for ), and

where

denotes the (-periodic) cross-covariance sequence between the sequences and in (see [29]). Thus, we deduce the following result.

Proposition 2. Any element can be recovered from the sequence of its samples in a stable way (by means of a frame expansion) if and only if the sequence is a frame for .

Having in mind that a tensor product of two sequences is a frame in the tensor product if and only if the respective factors are frames, we only need a characterization of the sequences and as frames for and , respectively. This study has been done in [3, 10], respectively.

(i) For the first one, consider the matrix of functions in

and its related constants

where denotes the transpose conjugate of the matrix , and (resp., ) the smallest (resp., the largest) eigenvalue of the positive semidefinite matrix . Observe that . Notice that in the definition of the matrix we are considering 1-periodic extensions of the involved functions , .

A complete characterization of the sequence as a frame or a Riesz basis for is given in the next lemma (see [10, Lemma ] or [11, Lemma ] for the proof).

Lemma 3. For the functions , , consider the associated matrix given in (22). Then, the following results hold:

(a)The sequence is a complete system for if and only if the rank of the matrix is a.e. in .(b)The sequence is a Bessel sequence for if and only if (or equivalently ). In this case, the optimal Bessel bound is .(c)The sequence is a frame for if and only if . In this case, the optimal frame bounds are and .(d)The sequence is a Riesz basis for if and only if it is a frame and .

(ii) For the second one, consider the matrix of cross-covariances

where each block , , is given by

In [3] the following lemma was proved.

Lemma 4. The sequence is a frame for (or equivalently, a spanning set since we are in finite dimension) for if and only if rank .

Notice that, necessarily, and and, consequently, the number of needed -systems must be .

3.1. The Sampling Result

In case that is a frame for we need to describe the family of appropriate (for sampling purposes) dual frames. In [3, 10] suitable dual frames of the frames for and for , respectively, have been obtained. For a notational and reading ease we recall them.

(i) For the first case, choose functions in , , such that

It was proven in [10] that the sequence is a dual frame of the frame in .

All the possible choices in (26) are given by the first row of the matrices given by

where denotes the Moore-Penrose pseudoinverse of given by

is any matrix with entries in , and is the identity matrix of order (see [30]). Notice that the entries of are essentially bounded in since the functions , , and are essentially bounded in .

(ii) For the second case, the -periodic extensions of the columns of a left-inverse of the matrix , written as

and constructed from a matrix such that

as in [3, Section  3.1], form an appropriate dual frame of in (see [3] for the details about the construction of ).

All the possible matrices satisfying (30) are given by the first rows of any left-inverse of the matrix . All these left-inverses can be expressed as

where denotes the Moore-Penrose pseudoinverse of and is any arbitrary matrix.

Finally, one deduce that the sequence is a dual frame of in . Indeed, we have the following lemma.

Lemma 5. Assume that and are dual frames in and and are dual frames in . Then, the sequences and form a pair of dual frames in the tensor product .

Proof. Let and , and , be the bounded operators defined by

The fact that and are dual frames in implies that each sequence , , is a bounded sequence (in fact, for every , where and are the upper frame bounds of and , resp.) which converges in the strong operator topology of to the identity operator . Thus, Lemma  2.3 in [26] yields that converges in the strong operator topology of to . Since

for every and every , , then, for every and every . Hence converges in the strong operator topology of to ; that is,

for every , which concludes the proof.

As a consequence, for each there exists a unique such that . This can be expressed as the frame expansion

Then, applying the isomorphism and the shifting property in Lemma 1 (here it is the point where we are using the fact that the proposed dual frames are convenient for sampling purposes) one gets

where , , and , .

Collecting the pieces we have obtained until now we prove the following result.

Theorem 6. Let , , , and let be the associated -system giving the samples of any as in (10), , . Assume that the function , , given in (19) belongs to or, equivalently, that for the associated matrix defined in (22). The following statements are equivalent:

(a) and rank .(b)There exist elements in the subspace , , , such that the sequence is a frame for , and for any the expansion

 holds.(c)There exists a frame for such that, for each , the expansion

 holds.

Proof. It only remains to prove that condition (c) implies condition (a). Indeed, applying the isomorphism to the expansion given in (c) one gets the frame expansion

Having in mind (18) and that the sequence is a Bessel sequence for , according to [8, Lemma  5.6.2], one gets that it is a frame for . In particular, this implies, via Lemmas 3 and 4, that and rank .

In cases and we are in the Riesz bases setting: see statement (d) in Lemma 3; moreover, the square matrix must be invertible (see [3, Corollary  4]). In fact, the following corollary holds.

Corollary 7. In addition to the hypotheses of the theorem above, assume that and . The following statements are equivalent:

(1) and the square matrix is invertible.(2)There exist unique elements , , , in the subspace such that the sequence is a Riesz basis for , and the expansion of any with respect to this basis is

In case that the equivalent conditions are satisfied, the vectors , , , satisfy the interpolation property

whenever , , , and .

Proof. The uniqueness of the expansion with respect to a Riesz basis gives the stated interpolation property (41).

A Representative Example. In consider the shift operator . Let be a function such that the sequence is a Riesz sequence in (e.g., the function may be a -spline). Consider also the Hilbert space of -periodic functions and, for a fixed , the unitary operator . Let be a function such that the set is linearly independent in ; obviously (e.g., choose a nonzero function taking the value outside the interval ).

In the tensor product consider the closed subspace , that is, the tensor product of the shift-invariant subspaces and in . Given the functions , , and , , and fixing the sampling periods and (recall that ), for each in we consider its samples defined by

where and (note that ).

Under the hypotheses in Theorem 6 there will exist functions , , and , , such that for any the sampling expansion (37) reads as follows:

In this particular example, adding some mild hypotheses we can also derive pointwise convergence in the above sampling formula. Indeed, assuming that the generators and are continuous functions on such that is bounded on and since is bounded on , it is easy to deduce that any function in is a continuous function defined by the pointwise sum

Besides, the subspace is a reproducing kernel Hilbert space (RKHS) since the evaluation functionals are bounded in . Namely, using Cauchy-Schwarz’s inequality in (44) and Riesz basis definition, for each , we have

where denotes the lower Riesz bound of the Riesz basis for . The above inequality shows that convergence in implies pointwise convergence which is uniform on . See, for instance, [31] for the relationship between sampling theory and RKHS.

4. Infinite-Infinite Generators Case

Let and be two separable Hilbert spaces and and two unitary operators. Consider two elements and such that the sequences and form a Riesz sequence in and in , respectively. In the tensor product Hilbert space we consider its closed subspace

Since the sequence is a Riesz basis for the tensor product of the -invariant subspace in and the -invariant subspace in we deduce that which can be described as

We will refer to the vectors as the infinite-infinite generators of the subspace in .

The Samples. For fixed elements , , , we consider two sampling periods . For each we introduce the sequence of its generalized samples

defined by

where , , and .

The Isomorphism . In this case we introduce the isomorphism which maps the orthonormal basis for the Hilbert space onto the Riesz basis for . That is, , where and denote the isomorphisms

Here, the shifting property reads as follows:

where and . The proof of (51) goes in the same manner as in Lemma 1.

An Expression for the Samples. For any in we have

Now, consider , which is the element in such that . Thus, Plancherel identity for orthonormal bases gives

That is, for , , and we have got the following expression for the samples:

where, for and , the functions

belong to . Thus, we deduce the following result.

Proposition 8. Any element can be recovered from the sequence of its samples in a stable way (by means of a frame expansion) if and only if the sequence is a frame for .

The thesis of the above proposition is true if and only if and , where and denote the and matrices defined in (22) for , , and , , respectively.

4.1. The Sampling Result

In case that is a frame for , a family of dual frames is given by , where the functions , , and , , in , satisfy

Thus, for each there exists a unique such that . This can be expressed in as the frame expansion

Then, applying the isomorphism and (51) one gets

where , , and , . Next, we state the equivalent result to Theorem 6; its proof goes in the same manner.

Theorem 9. Let , , and . Let be the associated -system giving the samples of any as in (49), and . Assume that the functions and , and , given in (55) belong to or, equivalently, that and for the correspondent and matrices and , respectively, defined in (22). The following statements are equivalent:

(a) and .(b)There exist elements in the subspace , and , such that the sequence is a frame for , and for any the expansion

 holds.(c)There exists a frame for such that, for each , the expansion

 holds.

In cases and we are in the Riesz bases setting and the theorem above admits the following corollary.

Corollary 10. In addition to the hypotheses of theorem above, assume that and . The following statements are equivalent:

(1) and .(2)There exist unique elements in the subspace , and , such that the sequence is a Riesz basis for , and the expansion of any with respect to this basis is

In case that the equivalent conditions are satisfied, the vectors , and , satisfy the interpolation property

whenever , , and .

A Representative Example. In consider the shift operator . Let be two functions such that the sequences and are Riesz sequences for (e.g., the functions and may be two -splines).

In the tensor product consider the closed subspace , that is, the tensor product of the shift-invariant subspaces and of . Given the functions , and , and fixing the sampling periods and in , for each in we consider its samples defined by

Under the hypotheses in Theorem 9 there will exist functions , , and , , such that for any the sampling expansion (59) reads as follows:

As in the example of the infinite-finite case, assuming that the generators and are continuous functions on such that the sums and are bounded on , it is easy to deduce that is a RKHS of continuous functions on . Furthermore, the convergence in implies pointwise convergence which is uniform on .

5. Finite-Finite Generators Case

Let and be two separable Hilbert spaces and and two unitary operators. Consider two elements and such that, for some , and and the sets and are linearly independent in and , respectively. In the tensor product Hilbert space we consider the finite subspace

The subspace coincides with the tensor product of the finite subspaces and . We will refer to the vectors as the finite-finite generators of the subspace in .

The Samples. For fixed elements , and , we consider two sampling periods, a divisor of and a divisor of ; denote and . For each we introduce its generalized samples

defined by

where , , , and .

The Isomorphism . As in the former cases, now we introduce the isomorphism which maps the orthonormal basis for the Hilbert space onto the basis for ; here and denote, respectively, the canonical bases for and . In other words, , where and denote the isomorphisms

Here, the shifting property reads as follows:

where

belong to and , and

belong to and . The proof of (69) goes in the same manner as in Lemma 1.

An Expression for the Samples. For any in we have

Now, consider , which is the element in such that . Thus, Plancherel identity for orthonormal bases gives

That is, for , , , and we have got the following expression for the samples:

where

Here, , , denotes the (-periodic) cross-covariance sequence between the sequences and in . Similarly, , , denotes the (-periodic) cross-covariance sequence between the sequences and in . Thus, we deduce the following result.

Proposition 11. Any element can be recovered from the sequence of its samples in a stable way (by means of a frame expansion) if and only if the sequence is a frame for .

The thesis of the above proposition is true if and only if the matrix (defined in (24) for ) has rank and the matrix (defined in (24) for ) has rank . Notice that, necessarily, and and .

5.1. The Sampling Result

In case that the sequence is a frame for , a family of appropriate dual frames is given by , where and are appropriate dual frames of and , respectively. These dual frames are constructed as in Section 3.1 (see [3, Section  3.1] for the details).

Thus, for each there exists a unique such that . This can be expressed as the frame expansion

Then, applying the isomorphism and (69) one gets

where , , and , . Next, we state the equivalent result to Theorem 6.

Theorem 12. Let , , , and let be the associated -system giving the samples of any as in (67), , . The following statements are equivalent:

(a)rank and rank .(b)There exist elements , and , in the subspace such that the sequence is a frame for , and for any the expansion

 holds.(c)There exists a frame for such that, for each , the expansion

 holds.

As in the previous cases, whenever and in the above theorem we are in the Riesz bases setting necessarily, and condition (a) says that both square matrices and are invertible.

A Representative Example. In consider the (circular) shift operator and which obviously satisfies and . Analogously, in consider the (circular) shift operator , and which obviously satisfies and .

Given the sequences , , and , , and fixing the sampling periods and (recall that and ), for each in , a double sequence in , -periodic in the first component, and -periodic in the second one, we consider its samples defined by

where , , , and .

Under the hypotheses in Theorem 12 there will be sequences , , and , , such that for any the sampling expansion (78) reads as follows:

where and .

6. Discussion and Conclusions

A sampling theory for tensor products of unitary invariant subspaces that allows merging the cases of finitely/infinitely generated unitary invariant subspaces formerly studied in the mathematical literature is derived. The involved samples are identified as frame coefficients in suitable tensor product spaces; thus it also allows introducing the several variables case in the formalism.

Alternatively, the theory developed here can also be considered as a theory of invariant subspaces in the tensor product of the corresponding Hilbert space with respect to the canonical unitary tensor product representation of the product group defined from the corresponding factors (in the case of infinite-finite generators).

In this sense, the results exhibited here point again in the direction recently stated in the paper [19] and natural generalizations of them, like sampling theorems for tensor products of invariant subspaces with respect to unitary representations of finite or infinite groups on each factor, can be addressed using similar ideas.

Reduction techniques corresponding to the decomposition of tensor products of unitary representations into their irreducible components with respect to proper subgroups of the product group could also be used to simplify the construction of samples, that is, to find adapted frames. This, as well as other applications beyond classical telecommunications all involving sampling of states of quantum systems, will be the subject of further research.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work has been supported by the Grant MTM2014-54692-P from the Spanish Ministerio de Economía y Competitividad (MINECO).

References

H. R. Fernández-Morales, A. G. García, M. A. Hernández-Medina, and M. J. Muñoz-Bouzo, “On some sampling-related frames in U-invariant spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 761620, 14 pages, 2013.

H. R. Fernández-Morales, A. G. García, M. A. Hernández-Medina, and M. J. Muñoz-Bouzo, “Generalized sampling: from shift-invariant to -invariant spaces,” Analysis and Applications, vol. 13, no. 3, pp. 303–329, 2015.

A. G. García and M. Muñoz-Bouzo, “Sampling-related frames in finite U-invariant subspaces,” Applied and Computational Harmonic Analysis, vol. 39, no. 1, pp. 173–184, 2015.

T. Michaeli, V. Pohl, and Y. C. Eldar, “U-invariant sampling: extrapolation and causal interpolation from generalized samples,” IEEE Transactions on Signal Processing, vol. 59, no. 5, pp. 2085–2100, 2011.

V. Pohl and H. Boche, “U-invariant sampling and reconstruction in atomic spaces with multiple generators,” IEEE Transactions on Signal Processing, vol. 60, no. 7, pp. 3506–3519, 2012.

A. Aldroubi, “Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces,” Applied and Computational Harmonic Analysis, vol. 13, no. 2, pp. 151–161, 2002.

A. Aldroubi and K. Gröchenig, “Nonuniform sampling and reconstruction in shift-invariant spaces,” SIAM Review, vol. 43, no. 4, pp. 585–620, 2001.

O. Christensen, An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, Mass, USA, 2003.

H. R. Fernández-Morales, A. G. García, and G. Pérez-Villalón, “Generalized sampling in shift-invariant subspaces with multiple stable generators,” in Multiscale Signal Analysis and Modeling, Lecture Notes in Electrical Engineering, pp. 51–80, Springer, New York, NY, USA, 2012.

A. G. García and G. Pérez-Villalón, “Dual frames in connected with generalized sampling in shift-invariant spaces,” Applied and Computational Harmonic Analysis, vol. 20, no. 3, pp. 422–433, 2006.

A. G. García, M. A. Hernández-Medina, and G. Pérez-Villalón, “Generalized sampling in shift-invariant spaces with multiple stable generators,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 69–84, 2008.

S. Kang and K. H. Kwon, “Generalized average sampling in shift invariant spaces,” Journal of Mathematical Analysis and Applications, vol. 377, no. 1, pp. 70–78, 2011.

C. V. M. van der Mee, M. Z. Nashed, and S. Seatzu, “Sampling expansions and interpolation in unitarily translation invariant reproducing kernel Hilbert spaces,” Advances in Computational Mathematics, vol. 19, no. 4, pp. 355–372, 2003.

Q. Sun, “Local reconstruction for sampling in shift-invariant spaces,” Advances in Computational Mathematics, vol. 32, no. 3, pp. 335–352, 2010.

W. Sun and X. Zhou, “Average sampling in spline subspaces,” Applied Mathematics Letters, vol. 15, no. 2, pp. 233–237, 2002.

W. Sun and X. Zhou, “Average sampling in shift invariant subspaces with symmetric averaging functions,” Journal of Mathematical Analysis and Applications, vol. 287, no. 1, pp. 279–295, 2003.

X. Zhou and W. Sun, “On the sampling theorem for wavelet subspaces,” The Journal of Fourier Analysis and Applications, vol. 5, no. 4, pp. 347–354, 1999.

H. R. Fernández-Morales, A. G. García, M. J. Muñoz-Bouzo, and A. Ortega, “Finite sampling in multiple generated U-invariant subspaces,” IEEE Transactions on Information Theory, vol. 62, no. 4, pp. 2203–2212, 2016.

A. G. García, A. Miguel, and A. Ibort, “Towards a quantum sampling theory: the case of finite groups,” https://arxiv.org/abs/1510.08134v1

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM, Philadelphia, Pa, USA, 2001.

H. Stark, “Polar, spiral, and generalized sampling and interpolation,” in Advanced topics in Shannon Sampling and Interpolation Theory, R. J. Marks II, Ed., pp. 185–207, Springer, New York, NY, USA, 1993.

R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic Fourier series,” Transactions of the American Mathematical Society, vol. 72, pp. 341–366, 1952.

I. Daubechies, A. Grossmann, and Y. Meyer, “Painless nonorthogonal expansions,” Journal of Mathematical Physics, vol. 27, no. 5, pp. 1271–1283, 1986.

P. G. Casazza and G. Kutyniok, Eds., Finite Frames: Theory and Applications, Birkhäuser, Boston, Mass, USA, 2014.

G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, Fla, USA, 1995.

A. Bourouihiya, “The tensor product of frames,” Sampling Theory in Signal and Image Processing, vol. 7, no. 1, pp. 65–76, 2008.

A. Khosravi and M. S. Asgari, “Frames and bases in tensor product of Hilbert spaces,” International Mathematical Journal, vol. 4, no. 6, pp. 527–537, 2003.

C. S. Kubrusly, “A concise introduction to tensor product,” Far East Journal of Mathematical Sciences, vol. 22, no. 2, pp. 137–174, 2006.

A. N. Kolmogorov, “Stationary sequences in Hilbert space,” Moscow University Mathematics Bulletin, vol. 2, pp. 1–40, 1941.

R. Penrose, “A generalized inverse for matrices,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 51, no. 3, pp. 406–413, 1955.

A. G. García, “Sampling theory and reproducing kernel Hilbert spaces,” in Operator Theory, D. Alpay, Ed., pp. 87–110, Springer, Basel, Switzerland, 2015.