Abstract and Applied Analysis

Volume 2013 (2013), Article ID 761620, 14 pages

http://dx.doi.org/10.1155/2013/761620

Research Article

On Some Sampling-Related Frames in -Invariant Spaces

H. R. Fernández-Morales,¹ A. G. García,¹ M. A. Hernández-Medina,² and M. J. Muñoz-Bouzo³

¹Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés-Madrid, Spain

²Departamento de Matemáticas Aplicada, E.T.S.I.T., U.P.M., Avda. Complutense 30, 28040 Madrid, Spain

³Departamento de Matemáticas Fundamentales, Facultad de Ciencias, U.N.E.D. Senda del Rey 9, 28040 Madrid, Spain

Received 16 May 2013; Accepted 19 September 2013

Academic Editor: Patricia J. Y. Wong

Copyright © 2013 H. R. Fernández-Morales 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.

This paper is concerned with the characterization as frames of some sequences in -invariant spaces of a separable Hilbert space where denotes an unitary operator defined on ; besides, the dual frames having the same form are also found. This general setting includes, in particular, shift-invariant or modulation-invariant subspaces in , where these frames are intimately related to the generalized sampling problem. We also deal with some related perturbation problems. In doing so, we need the unitary operator to belong to a continuous group of unitary operators.

1. Introduction

This paper is concerned with the study of some special frames in -invariant spaces. Given an unitary operator on a separable Hilbert space , we consider closed subspaces having the form , where denotes some fixed element in . In case that the sequence is a Riesz basis for , we have

Recall that a Riesz basis in a separable Hilbert space is the image of an orthonormal basis by means of a bounded invertible operator. Any Riesz basis has a unique biorthogonal (dual) Riesz basis , that is, , such that the expansions

hold for every . We state the definition by considering the integers set as the index set since throughout the paper most of sequences are indexed in . A Riesz sequence in is a Riesz basis for its closed span. A necessary and sufficient condition in order for the sequence to be a Riesz sequence in is given in Theorem 3 infra.

Given elements , , in , a challenging problem is to characterize the sequence as a frame (Riesz basis) in , where denotes a positive integer. Besides, another interesting problem is to look for those dual frames having the same form for some , , so that, for any the expansion

holds.

At this point, we give an explanation about the expression sampling-related frames appearing in the title. Namely, -invariant subspaces in are natural generalizations of shift-invariant or modulation-invariant subspaces of ; there, the unitary involved operators are, respectively, the translation operator or the modulation operator . In the shift-invariant subspace generated by , for any , the inner products are

where for each . Thus, the above inner products are nothing but samples of some filtered versions of the function itself; this is precisely the generalized sampling problem in the shift-invariant space . Mathematically, it consists of the stable recovery of any from the above sequence of samples, that is, to obtain sampling formulas in having the following form:

such that the sequence of reconstruction functions is a frame for the shift-invariant space . As a consequence, expansions (3) and (5) have the same nature. Recall that a sequence is a frame for a separable Hilbert space if there exist constants (frame bounds), such that

A sequence in satisfying only the right hand inequality above 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 (exact frames), the uniqueness of this representation (for overcomplete frames) is sacrificed. Suitable frame coefficients which depend continuously and linearly on are obtained by using the dual frames of ; that is, is another frame for such that for each . For more details on the frame theory see [1].

Sampling in shift-invariant spaces of has been profusely treated in the mathematical literature (see, for instance, [2–13]).

The existence of expansions like (3) in -invariant subspaces was treated for the first time in [14]; see also [15, 16]. Following similar techniques to those in [14], we give, in Section 2, a complete characterization in of sequences having the form where for each . In other words, we carry out the study of the completeness, Bessel, frame, or Riesz basis properties of the sequence . Whenever it is a frame for , we find a family of dual frames having the same form for some , . In Section 4, we also discuss the case where some ; although the sequence is not contained in , something can be said in the light of the theory of pseudoframes (see [17, 18]).

All the obtained results in Section 2 involve the discrete group of unitary operators which is completely determined by . If we want to deal with something similar to the time-jitter error version of (5), that is, the recovery of any from a perturbed sequence of samples with errors , then the availability of a continuous group of unitary operators containing, in particular, the operator (say for instance ) becomes essential. In Section 3, after a brief on groups of unitary operators, we deal with two types of perturbation problems. The first one concerns the study of sequences as in , for small enough error sequence -norm, the sequence is a Riesz sequence in . The second one goes into the recovery of any from the perturbed sequence of inner products , for small enough errors , there exists a frame expansion for having the inner products as coefficients.

2. -Invariant Subspaces

In a Hilbert space , the -invariant subspaces are intimately related to stationary sequences.

2.1. Some Preliminaries on Stationary Sequences

A sequence in a separable Hilbert space is said to be stationary if

The function , for every , is called the autocovariance function of the sequence . Moreover, two stationary sequences and are said to be stationary correlated if

and , for every defines the corresponding cross-covariance function. The following result is a well-known characterization of stationary sequences (see [19]).

Lemma 1. To every stationary sequence in a Hilbert space there exists a unique unitary operator and such that for all . Conversely, every pair of a unitary operator and an defines by , , a stationary sequence in .

Moreover, two stationary sequence and are stationary correlated if and only if they are generated by the same unitary operator ; that is, and for some .

The autocovariance and the cross-covariance functions admit a spectral representation which is related to the integral representation of the unitary operator (see [19]).

Theorem 2. For every stationary sequence in a Hilbert space , the autocovariance function admits a spectral representation

in the form of an integral with respect to a (positive) spectral measure .

For every two stationary correlated sequences , in a Hilbert space , the cross-covariance function admits a spectral representation

in the form of an integral with respect to a (complex) spectral measure .

2.1.1. The -Invariant Subspace

For a fixed , we consider the subspace of given by . In case that the sequence is a Riesz sequence in , we have

A necessary and sufficient condition in order for the sequence to be a Riesz sequence in can be stated in terms of the Lebesgue decomposition of the spectral measure into an absolute and a singular part as .

Theorem 3. Let be a sequence obtained from an unitary operator in a separable Hilbert space with spectral measure , and let be the closed subspace spanned by . Then, is a Riesz basis for if and only if and

where .

Proof. For a fixed -sequence , we have

if is not absolutely continuous with respect to Lebesgue measure , Lemma 4 below implies that there exists a bounded sequence such that tends to infinity with , so cannot be a Bessel sequence, therefore, not a Riesz basis. Assume now that and is a Riesz basis, this implies that there exists such that

for all . Let , assume that , this implies that for all . Now, (13), (14), and Parseval's theorem imply that

for all . Introducing the Fourier expansion of the characteristic function in (15), we obtain for all , which contradicts ; thus, the assumption is false. In a similar way, it can be proved that .

For the sufficient condition, assume now that ; then, (13) implies that

if, in addition, condition (12) is satisfied, we get

which implies that

Therefore, the sequence is a Riesz basis for .

Lemma 4. Let be a finite positive measure on which is not absolutely continuous with respect to Lebesgue measure . Then, there exists a bounded sequence such that

Proof. If the measure is not absolutely continuous with respect to Lebesgue measure, then for a Lebesgue measurable set of Lebesgue measure zero. Thus, there exists a Borel set of Lebesgue measure zero such that . In fact, the set is an intersection of a countable collection of open sets (see [20, page 63]). Therefore, and . On the other hand, every finite Borel measure on is inner regular (see [20, page 340]); that is,

then there exists a compact set such that

For any , there exists a sequence of disjoint open intervals such that

(see [20, pages 58 and 42]). Since is compact, we may take the sequence to be finite. Hence, for every , there exist open disjoint intervals in such that

Besides, .

Consider the function , where , that satisfies

We modify and extend each to obtain a -periodic function such that and its derivative are continuous on , and for every . Let be the Fourier series of . First, by using Parseval's identity, we have

so that is a bounded sequence in . Besides, the regularity of each ensures that each Fourier series converges uniformly to . Therefore, each series converges to in and consequently,

from which we obtain the desired result.

The proof of Theorem 3 is similar to that of Theorem 6 in [16], except we do not exclude the case in which the singular measure is atomless. Recently, we have been aware that Theorem 3 was exposed in [21].

2.2. Studying the Sequence in

For , , consider the sequence . For every , the spectral measure in the integral representation of the cross-covariance function of the sequences and has no singular part. Indeed, according to Theorem 3, the spectral measure associated with the autocovariance function of the sequence has no singular part; then by using the Cauchy-Schwarz type inequality in [22, page 125] we get the result. In the sequel, we will use the abridged notation ; our goal in this section is to study the sequence in in terms of an matrix which is introduced below. For the sake of completeness, we include some needed calculations which appear in [14].

First of all, we have

where stands for the cross spectral density of the stationary correlated sequences and . Define

In what follows, we will use the left-shift operator defined as

or equivalently, by , where denotes the unidimensional torus. Also, we will consider the decimation operator , is a positive integer, defined as

which can equivalently be written as

For each , set the matrix of functions on the torus as follows:

and define the matrix of functions on the torus as follows:

It is worth to mention that the matrix was explicitly computed in [14] for the translation and modulation cases in .

Next, for any , we obtain an expression for the inner products , and . Indeed, writing where , we have

that is,

where .

Now, for , define the sequence . Thus, we can write

where .

Using (36) in (35), we obtain

After some easy calculations, we get

Defining , (38) implies that

which can be written in matrix form as

where denotes the multiplication operator by and

By (resp., ), we denote the product space for times (resp., times). Besides,

The above calculations let us prove the following result.

Theorem 5. Let for and let be the associated matrix given in (33). Then, the following results hold.

(a)The sequence is a complete system in 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 there exists a constant such that

(c)The sequence is a frame for if and only if there exist constants such that

(d)The sequence is a Riesz basis for if and only if it is a frame and .

Proof. To prove (a), assume that there exists a set with positive measure such that for each . Then, there exists a measurable function , , such that and in . This function can be constructed as in [23, Lemma 2.4]. Define such that if , and if . Hence, from (40), we obtain that the system is not complete. Conversely, if the system is not complete, by using (40), we obtain a different from 0 in a set with positive measure such that . Thus, on a set with positive measure.

To prove (b), we keep in mind that is a Riesz basis for , the mapping , given by is bijective and there exist two constants such that

Assume first that (43) is satisfied. It follows from (40) and (42) that

By construction and , using (45), it follows from (46) that

Conversely, assume that is a Bessel sequence for , then there exists such that

Using (45), this implies that

for all . Inserting the right hand side of (42) for , it is straightforward to see that (43) holds with . The proof of (c) is completed proceeding as in (b).

To prove (d), consider the following mapping:

According to (40), the mapping is isometric equivalent to , and assuming that is a frame, it is a Riesz basis if and only if is surjective.

First, if is a Riesz basis, then it is a frame and is surjective. Applying (a) yields that is bijective, and therefore is bijective. Hence, is for almost every in , so

and finally .

Conversely, if is a frame and , (a) implies that is invertible for almost every in , which implies that is surjective; then is surjective and is a Riesz basis.

The following lemma will allow us to restate Theorem 5.

Lemma 6. Let be an matrix with entries in , and consider the following constants:

where (resp., ) denotes the smallest (resp., the largest) eigenvalue of the positive semidefinite matrix . Then,

(a)the matrix has essentially bounded entries on if and only if ;(b)there exist constants such that , a.e., if and only if .

Proof. The first part of lemma follows from that , and

where denotes the spectral norm of the matrix (see, for instance, [24]).

Now, we prove the second part of the lemma. Since means that for all , in particular, taking an eigenvector associated with the largest eigenvalue of such that , one has that . Hence, . In a similar way, implies that .

Conversely, Rayleigh-Ritz theorem [24, page 176] yields that

Thus, implies that

In other words, ; analogously, .

It is easy to deduce from the proof that and are the optimal constants and satisfying the inequalities , a.e., .

As a consequence of Lemma 6, statements (b) and (c) in Theorem 5 can be restated in terms of the following constants:

as follows.

Theorem 7. Let for , and let be the associated matrix given in (33) and its related constants (56). Then, the following results hold.

(i)The sequence is a Bessel sequence for if and only if the constant . (ii)The sequence is a frame for if and only if the constants and satisfy . In this case, and are the optimal frame bounds for .

2.3. The Frame Expansion

Define the matrix of functions on the torus as follows:

The following expansion involving the inner products of holds.

Lemma 8. Assume that the matrix has essentially bounded entries on . For any , one has

where denotes the th Fourier coefficient of the function defined in (41), and the sequence is given in (35).

Proof. Indeed,

At this point, we are ready to prove the following expansion result.

Theorem 9. Let for , and assume that the associated matrix given in (33) has essentially bounded entries on , that is, . The following statements are equivalent.

(i)The constant . (ii)There exist , , such that the sequence is a frame for , yielding, for any , the following expansion:

In case the equivalent conditions hold, and form a pair of dual frames in .

Proof. First, we prove that (i) implies (ii). Observe that can be written as where . Next,

For and define , where , and . Then, (61) implies

In order to be allowed to use [1, Lemma 5.6.2], we have to prove that the above constructed sequence is a Bessel sequence for . To this end, we compute the corresponding matrix for . Setting , we obtain

Now,

where . Hence, we have deduced that

Therefore, for , we have

and consequently, the matrix can be written as

where

As a consequence of Theorem 7, the proof ends if we prove that the matrix has essentially bounded entries. Clearly, the decimation operator sends bounded functions into bounded functions; Theorem 3 implies that is bounded so, taking into account (67), it remains to check that the matrix has essentially bounded entries.

Now, , the lower bound condition (c) in Theorem 5 and Lemma 6 imply that has bounded entries, and therefore the matrix has bounded entries. We have shown that has bounded entries; then Theorem 7, part (a) and Lemma 6 guarantee that the sequence is a Bessel sequence; then, the sequences and form a pair of dual frames in (see [1, Lemma 5.6.2]).

Finally, condition (ii) implies condition (i). According to [1, Lemma 5.6.2], the sequence is a frame for since it is a Bessel sequence and the expansion in (ii) holds. By using Theorem 7 we obtain that .

It is worth to observe that the analysis done in Theorem 9 provides a whole family of dual frames for the sequence . In fact, everything works if we replace in (57) by any matrix of the following form:

where is any matrix with entries in , and denotes the Moore-Penrose left-inverse . Note that we need essentially bounded entries in the matrix since the multiplication operator in is well-defined (and consequently bounded) if and only if .

Notice that if , we have which implies a unique , and we are in the presence of a pair of dual Riesz bases. In fact, the following result holds.

Corollary 10. Let for ; that is, in Theorem 9. Assume that the square matrix given in (33) has entries essentially bounded on , that is, . The following statements are equivalent.

(a) The constant . (b) There exists a Riesz basis such that for any , the expansion

 holds.

In case the equivalent conditions are satisfied, necessarily, there exist , , such that for and . Moreover, the sequences and are dual Riesz bases in , and the interpolation property , where and , holds.

Proof. To prove , we use Theorem 9; whenever , there exist , , such that the sequence is a frame for and for any , the expansion

holds. Actually, from Theorem 5, we get that implies that is a Riesz basis, and consequently, is indeed its dual Riesz basis.

The converse follows easily from the fact that if is a Riesz basis, then (b) implies that is its dual Riesz basis; hence, Theorem 5 provides . The interpolation property comes out from the biorthogonal condition of a pair of dual Riesz bases.

Closing this section it is worth to mention that the results stated and proved in Sections 2.2 and 2.3 mathematically enrich some of the remarkable results concerning regular sampling contained in the interesting [14]. Here, we have assumed only one generator and that for all . If for some , see the additional remarks in Section 4. The case of several generators , , can be essentially treated in the same way.

3. Some Perturbation Results

In Section 2, we have only used the discrete group of unitary operators which is completely determined by the unitary operator . In order to carry out the study of some perturbation results stated below, we need the availability of a continuous group of unitary operators which includes the unitary operator , say, for instance, that .

3.1. On Continuous Groups of Unitary Operators

Let denote a continuous group of unitary operators in such that . Recall that is a family of unitary operators in satisfying (see [25, vol. 2; page 29]) the following:

(1),(2),(3) is a continuous function of for any .

Note that , and since , we have .

Classical Stone's theorem [26] assures us the existence of a self-adjoint operator (possibly unbounded) such that . This self-adjoint operator , defined on the dense domain of as follows:

admits the spectral representation   which means that

where is the corresponding resolution of the identity, that is, a one-parameter family of projection operators in such that

(i), , (ii) for every , (iii) where .

Recall that and , as functions of , have bounded variation and define, respectively, a positive and a complex Borel measure on .

Furthermore, for any , we have that and the operator is said to be the infinitesimal generator of the group . For each , is a continuous differentiable function of . Notice that, whenever the self-adjoint operator is bounded, and can be defined as the usual exponential series; in any case, means that

where and . A comment on the continuity of a group of unitary operators is in order: The group is said to be strongly continuous if, for each and , as . If is a separable Hilbert space, strong continuity can be deduced from continuity and even from weak measurability; that is, is a Lebesgue measurable function of for any .

The following result taken from [25, vol. 2; page 24] will be used later. For and , the inequality

holds, where is a real or complex function which is continuous in with the possible exception of a finite number of points.

For the details on the theory of continuous groups of unitary operators, see [22, 25, 27, 28].

3.2. Studying the Perturbed Sequence

Given an error sequence , consider the corresponding perturbed sequence . Consider the following perturbation result (see [1, page 354] for the proof).

Lemma 11. Let be a frame for the Hilbert space with frame bounds , and let be a sequence in . If there exists a constant such that

then, the sequence is also a frame for with bounds and . If is a Riesz basis, then is a Riesz basis.

Note that it cannot be directly applied to the sequences and since the first one is not a frame for the entire Hilbert space , and its perturbed sequence does not necessarily belong to the subspace . However, something can be said in case is a Riesz sequence in .

Theorem 12. Assume that for some ; that is, for each , the sequence is a Riesz basis for with Riesz bounds . For a sequence of errors, let be the constant given by

where denotes the -norm of the sequence .

If , then the sequence is a Riesz sequence in with Riesz bounds and .

Proof. By using (75), we have

Hence,

Hence, Lemma 11 and Theorem 15.3.2 in [1, page 356] give the desired results.

3.3. On the Perturbed Frame Expansion

Next, we deal with the problem of the recovery of any in a stable way from the following perturbed sequence:

where denotes a sequence of real errors. In order to face this problem, we propose a possible strategy. Let be the isomorphism which maps the orthonormal basis onto the Riesz basis for . In other words,

Thus,

where , and the function belongs to since the sequence belongs to for each .

Hence, for any , we have the following expressions:

where the functions

belong to . Therefore, we can see the sequence as a perturbation of the sequence . From [6, Lemma 3], we know that this sequence is a frame for if and only if where the constants and are given by

and is the matrix

Besides, the optimal frame bounds for are and .

Given an error sequence , we define

For any sequence , we have

Now, for , define the following functions:

Notice that and for , the equality holds. Moreover, assuming that the continuous functions , , satisfy a decay condition as when for some , we may deduce that the functions and are continuous near to .

Theorem 13. Assume that for the functions , , given in (84) one has . For an error sequence , define the constant for each . Then the following condition:

implies that there exists a frame for such that, for any , the following sampling expansion:

holds. Moreover, when , the sequence is a Riesz basis for , and the interpolation property holds.

Proof. The sequence is a frame (a Riesz basis if ) for with frame (Riesz) bounds and . For any in , we have

From (88) and the definition of the functions and , we obtain

By using Lemma 11, we obtain that the sequence is a frame for (a Riesz basis if ). Let be its canonical dual frame. Hence, for any ,

Applying the isomorphism , one gets (91), where . Since is an isomorphism between and ; the sequence is a frame for (a Riesz basis if ). The interpolatory property in the case follows from the uniqueness of the coefficients with respect to a Riesz basis.

3.4. A Frame Algorithm in

Sampling formula (91) is useless from a practical point of view, it is impossible to determine the involved frame . As a consequence, in order to recover from the sequence of inner products , we should implement a frame algorithm in . Another possibility is given in the recent reference [29].

Next we are going to implement a frame algorithm in the setting. To this end, consider the canonical isometry

For , denote by the following sequence:

The inner products can be written as

where, for and ,

The sequence is a frame for . Indeed, assume that the error sequence satisfies the hypothesis of Theorem 13; that is,

where for each . As a consequence, the sequence is a frame for with bounds

Since is an isometry, the sequence is a frame for with the same bounds. Hence, the recovery of the element from the samples is reduced to recover from the following sequence:

In doing so, the classical frame algorithm reads (see, for instance, [1]), let be the frame operator in of , and define

and recursively,

Then, the sequence in given by , satisfies

where .

4. Some Additional Remarks

Given vectors , with , we have proved in Section 2 that the sequence is a frame for if and only if the constants and defined in (56) satisfy . Furthermore, we have obtained a family of dual frames having the same form. As it was mentioned in the introduction, now we deal with the case that some .

(i) We have assumed in Theorems 5 and 7 that belongs to for each since we required the sequence to be contained in . In case that some , the sequence is not necessarily contained in . However, a close look into the proof of Theorem 5 shows that whenever , the following inequalities:

 hold, and conversely. Hence, the sequence is a pseudoframe for (see [17, 18]).

Denoting by the orthogonal projection onto , since for each ,

Theorems 5 and 7 can be reformulated in terms of which is a sequence in .

(ii) An analysis of the proof of Theorem 9 shows that, even if not all of the belong to , there exist , , such that the sequence is a frame for , and the expansion (60) holds for each . Therefore, in case that some , the sequence is a pseudodual frame of the frame for (see [17]).(iii) In Section 3.3, having in mind the isomorphism , for , we have obtained the following expressions:

Furthermore, we know that the sequence is a frame for if and only if the constants and given in (85) satisfy ; besides, the optimal frame bounds for are and . Hence, we obtain

As a consequence, since we are dealing with optimal frame bounds, from (105) and (108), we derive the following equalities:

Acknowledgments

The authors are indebted to the anonymous referee for her/his valuable suggestions. This work has been supported by Grant MTM2009-08345 from the Spanish Ministerio de Ciencia e Innovación (MICINN).

References

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

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.

I. Djokovic and P. P. Vaidyanathan, “Generalized sampling theorems in multiresolution subspaces,” IEEE Transactions on Signal Processing, vol. 45, no. 3, pp. 583–599, 1997.

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, 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.

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.

A. Papoulis, “Generalized sampling expansion,” IEEE Transactions on Circuits and Systems, vol. 24, no. 11, pp. 652–654, 1977.

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.

M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Processing Magazine, vol. 16, no. 6, pp. 22–38, 1999.

M. Unser, “Sampling: 50 years after Shannon,” Proceedings of the IEEE, vol. 88, no. 4, pp. 569–587, 2000.

M. Unser and A. Aldroubi, “General sampling theory for nonideal acquisition devices,” IEEE Transactions on Signal Processing, vol. 42, no. 11, pp. 2915–2925, 1994.

G. G. Walter, “A sampling theorem for wavelet subspaces,” IEEE Transactions on Information Theory, vol. 38, no. 2, pp. 881–884, 1992.

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

H. R. Fernández-Morales, A. G. García, M. A. Hernández-Medina, and M. J. Muñoz-Bouzo, “Generalized sampling: from shift-invariant to -invariant spaces,” Submitted

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

S. Li and H. Ogawa, “Pseudo-duals of frames with applications,” Applied and Computational Harmonic Analysis, vol. 11, no. 2, pp. 289–304, 2001.

S. Li and H. Ogawa, “Pseudoframes for subspaces with applications,” The Journal of Fourier Analysis and Applications, vol. 10, no. 4, pp. 409–431, 2004.

A. N. Kolmogorov, Stationary Sequences in Hilbert Space, vol. 2, Moscow University Mathematics Bulletin, 1941.

H. L. Royden, Real Analysis, Macmillan Publishing Company, New York, NY, USA, 3rd edition, 1988.

P. Gavruta, “Wavelets and spectral measures,” in Proceedings of the International Workshop on Operator Theory and its Applications (IWOTA ’00), Bordeaux, France, 2000.

M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, Springer, New York, NY, USA, 1987.

R. Q. Jia and Z. Shen, “Multiresolution and wavelets,” Proceedings of the Edinburgh Mathematical Society, vol. 37, no. 2, pp. 271–300, 1994.

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1999.

N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover Publications, New York, NY, USA, 1993.

M. H. Stone, “On one-parameter unitary groups in Hilbert space,” Annals of Mathematics, vol. 33, no. 3, pp. 643–648, 1932.

J. Weidmann, Linear Ooperators in Hilbert Spaces, vol. 68, Springer, New York, NY, USA, 1980.

K. Yosida, Functional Analysis, Springer, New York, NY, USA, 1995.

B. Adcock and A. C. Hansen, “Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon,” Applied and Computational Harmonic Analysis, vol. 32, no. 3, pp. 357–388, 2012.