Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 163814, 7 pages

http://dx.doi.org/10.1155/2013/163814

Research Article

Construction of Frames for Shift-Invariant Spaces

Stevan Pilipović¹ and Suzana Simić²

¹Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovica 4, 21000 Novi Sad, Serbia

²Department of Mathematics and Informatics, Faculty of Science, University of Kragujevac, Radoja Domanovića 12, 34000 Kragujevac, Serbia

Received 9 May 2013; Accepted 25 July 2013

Academic Editor: Satit Saejung

Copyright © 2013 Stevan Pilipović and Suzana Simić. 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.

We construct a sequence which constitutes a -frame for the weighted shift-invariant space , , and generates a closed shift-invariant subspace of . The first construction is obtained by choosing functions , , with compactly supported Fourier transforms , . The second construction, with compactly supported , gives the Riesz basis.

1. Introduction and Preliminaries

The shift-invariant spaces , , quoted in the abstract, are used in the wavelet analysis, approximation theory, sampling theory, and so forth. They have been extensively studied by many authors [1–18]. The aim of this paper is to construct , , spaces with specially chosen functions , , which generate its -frame. These results extend and correct the construction obtained in [19]. For the first construction, we take functions , , so that the Fourier transforms are compactly supported smooth functions. Also, we derive conditions for the collection to form a Riesz basis for . We note that the properties of the constructed frame guarantee the feasibility of a stable and continuous reconstruction algorithm in [20]. We generalize these results for a shift-invariant subspace of . The second construction is obtained by choosing compactly supported functions , . In this way, we obtain the Riesz basis.

This paper is organized as follows. In Section 2 we quote some basic properties of certain subspaces of the weighted and spaces. In Section 3 we derive conditions for functions of the form , , , , to form a Riesz basis for . We also show that using functions of the form , , where is compactly supported smooth function whose length of support is less than or equal to , we cannot construct a -frame for the shift-invariant space . In Section 4 we construct a sequence , where or , which constitutes a -frame for the weighted shift-invariant space . Our construction shows that the sampling and reconstruction problem in the shift-invariant spaces is robust in the sense of [1]. In Section 5 we construct -Riesz basis by using compactly supported functions , .

2. Basic Spaces

Let a function be nonnegative, continuous, symmetric, and submultiplicative; that is, , ; let a function be -moderate; that is, , . Functions and are called weights. We consider the weighted function spaces and the weighted sequence spaces with -moderate weights (see [19]). Let . Then (with obvious modification for )

In what follows, we use the notation . Define , where , or , . With we denote the Fourier transform of the function ; that is, , .

Let and , . We define, as in [1], the semiconvolution as , , and .

The concept of a -frame is introduced in [1].

It is said that a collection is a -frame for if there exists a positive constant (dependent upon , , and ) such that

Recall [21] that the shift-invariant spaces are defined by

Remark 1 (see [22]). Let and let be -moderate. Then is a subspace (not necessarily closed) of and for any . Clearly (2) implies that and are isomorphic Banach spaces.

Let . Let

where we assume that is integrable for any , . Let be an matrix and . Then .

We recall results from [1, 19] which are needed in the sequel.

Lemma 2 (see [1]). The following statements are equivalent.

(1) is a constant function on . (2) is a constant function on . (3)There exists a positive constant independent of such that

The next theorem [19] derives necessary and sufficient conditions for an indexed family to constitute a -frame for , which is equivalent with the closedness of this space in . Thus, it is shown that under appropriate conditions on the frame vectors, there is an equivalence between the concept of -frames, Banach frames, and the closedness of the space they generate.

Theorem 3 (see [19]). Let , , and let be -moderate. The following statements are equivalent.

(i) is closed in .(ii) is a -frame for .(iii)There exists a positive constant such that

(iv)There exist positive constants and (depending on and ) such that

(v)There exists , such that

Corollary 4 (see [19]). Let , , and let be -moderate.

(i)If is a -frame for , then the collection is a -frame for for any .(ii)If is closed in and , then is closed in and for any .(iii)If (7) holds for , then it holds for any .

3. Construction of Frames Using a Band-Limited Function

Considering the length of the support of a function , we have different cases for the rank of matrix .

First, we consider the next claim.

Let be a nonnegative function such that , , and . Moreover, let

and , , .

Then the rank of matrix is not a constant function on and it depends on .

As a matter of fact, by the Paley-Wiener theorem, , . For any , matrix , , has the rank or , depending on . Moreover, we have and . Because of that, the rank of the matrix is not a constant function on and it depends on .

Theorem 5. Let be a non-negative function such that , , and , where . Moreover, let

and .

(1)If and for different , then

(2)If and, at least for and , it holds that , where , , then is not a constant function on .

Proof.  By the Paley-Wiener theorem, , . All possible cases are described in the following lemmas.

Lemma 6. Let , , . The rank of matrix is a constant function on and equals 2.

Proof. We have the next two cases.

If , , for matrix we obtain matrix

for some , . It is obvious that , , .

For , , there are only two nonzero values and which are in different columns of matrix . Since

it has the rank 2 for all , .

We conclude that the rank of matrix , , , , is a constant function on and equals 2.

Lemma 7. The rank of matrix is not a constant function on if , , .

Proof. We have four different cases for matrix . Suppose, without losing generality, that .

If , , then

and , for all , .

For , , nonzero values and are in the same column of matrix . For any choice of a 2 × 2 matrix, we get that the determinant equals 0. So we obtain

for all , .

If , , then

for some , has the rank , for all , .

For , , there are two non-zero values and in different columns of matrix and the block with these elements determines the rank 2 for all , .

Considering possible cases, we conclude that , , , , depends on and equals 1 or 2. This rank is a nonconstant function on .

Proof of Theorem 5. (1) Using Lemmas 6 and 7, it is obvious that if and for different , then the position of the first non-zero element in each row of matrix is unique for each row. So the rank of matrix is a constant function on and equals for all .

(2) If and, at least for and , it holds that , , , then, in the row with the index (suppose, without losing generality, that ), we will have a new column with a non-zero element for , , but for , , the positions of all non-zero elements in that row will appear in the previous columns. It is obvious that the rank of the matrix depends on and is not the same for all .

As a consequence of Theorems 3 and 5(1), we have the following result.

Theorem 8. Let the functions and satisfy all the conditions of Theorem 5(1). Then space is closed in for any , and the family is a -Riesz basis for for any .

The following theorem is a generalisation of Theorem 5 and can be proved in the same way, so we omit the proof.

Theorem 9. Let be a positive function such that , , , and supported by where . Moreover, let

and .

(1)If and for different , then

(2)If and, at least for and , it holds that , where , , then is not a constant function on .

4. Construction of Frames Using Several Band-Limited Functions

Firstly, we consider two smooth functions with proper compact supports.

Lemma 10. Let , be positive functions such that

Moreover, let , , , and . Then , .

Proof. Note that , .

We have the following two cases.

If , , then matrix

has a constant rank equal to .

For , , the rank of matrix

where , are non-zero values, is equal to 1. An equivalent matrix is obtained for and , so we conclude that , for , .

Considering these two cases, the rank of matrix is a constant function on , , .

Using functions and from Lemma 10, in the next lemma we construct the -frame with four functions.

Lemma 11. Let the functions and satisfy all the conditions of Lemma 10. Moreover, let

and . Then , .

Proof. The proof is similar to the proof of Lemma 10.

If , , then matrix

where , , has a constant rank equal to .

For , , we have

where , .

We conclude that the rank of matrix is a constant function on equal to 2.

Lemma 10 can be easily generalised for an even number of functions , , with compactly supported , . The proof of the next theorem is similar to the previous proofs.

Theorem 12. Let the functions and satisfy all the conditions of Lemma 10. Moreover, let

and .

The following statements hold.

(1°) for all .(2°) is closed in for any .(3°) is a -frame for for any .

Now we consider three functions with compact supports.

Lemma 13. Let the function satisfies all the conditions of Lemma 10, and let and be positive functions such that

Moreover, let , , , , and . Then , .

Proof. We have four different forms for matrix and in all we show that the rank of matrix is equal to .

Now we will show all possible cases. Denote with , , and , , some positive values.

Consider

Consider

Consider

Consider

Remark 14. In Lemma 13 the support of the function must have an empty intersection with the supports of and . In the opposite case, that is, , the rank of the matrix is a non-constant function on .

Lemma 13 can be easily generalised for functions , , with compactly supported , . The proof of the next theorem is similar to the previous proofs.

Theorem 15. Let the functions , , and satisfy all the conditions of Lemma 13. Moreover, let

, , and .

The following statements hold.

(1°) for all .(2°) is closed in for any .(3°) is a -frame for for any .

5. Construction of Frames of Functions with Finite Regularities and Compact Supports

We will recall the well-known construction of the -spline functions in order to justify the rank properties of the corresponding matrices.

Let , , be the characteristic function of the semiaxis ; that is, if and if (Heaviside’s function). We construct a sequence in the following way. Let , , , , ; that is,

where denotes the convolution of the functions.

We obtain

Continuing in this manner, for all , we have

Calculating the Fourier transform of functions , , we get

Continuing in this manner, we obtain , , where denotes the principal value.

Let , . matrix has for all the same rank as matrix

where and . Since , , we have the next result.

Theorem 16. Let , for , . Then is closed in for any and is a -Riesz basis for for any .

Remark 17. Let be a positive integer. We refer to [23] for the -approximation order . Shift-invariant spaces generated by a finite number of compactly supported functions in , , were studied in [23] by Jia, who gave a characterization of the approximation order providing such shift-invariant spaces. Theorem  3 in [23] shows that the shift-invariant space generated with the family of splines, which we constructed in Section 5, provides -approximation order .

Remark 18. (1) We refer to [3, 20] for the -dense set . Let , . Following the notation of [20], we put , where is -dense set determined by . Theorems 3.1, 3.2, and  4.1 in [20] give conditions and explicit form of and such that inequality holds. This inequality guarantees the feasibility of a stable and continuous reconstruction algorithm in the signal spaces [20].

(2) Since the spectrum of the Gram matrix , where is defined in Theorem 16, is bounded and bounded away from zero (see [7]), it follows that the family forms a -Riesz basis for .

(3) Frames of the above sections may be useful in applications since they satisfy assumptions of Theorems 3.1 and 3.2 in [4]. They show that error analysis for sampling and reconstruction can be tolerated or that the sampling and reconstruction problem in shift-invariant space is robust with respect to appropriate set of functions .

Acknowledgments

The authors are indebted to the referee for the valuable suggestions, which have contributed to the improvement of the presentation of the paper. The authors were supported in part by the Serbian Ministry of Science and Technological Developments (Project no. 174024).

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