Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 757039 | https://doi.org/10.1155/2010/757039

Liu Zhanwei, Hu Guoen, Wu Guochang, "Regular and Irregular Sampling Theorem for Multiwavelet Subspaces", Mathematical Problems in Engineering, vol. 2010, Article ID 757039, 18 pages, 2010. https://doi.org/10.1155/2010/757039

Regular and Irregular Sampling Theorem for Multiwavelet Subspaces

Academic Editor: Victoria Vampa
Received05 Mar 2010
Accepted26 Jul 2010
Published20 Sep 2010

Abstract

We study the sampling theorem for frames in multiwavelet subspaces. Firstly, a sufficient condition under which the regular sampling theorem holds is established. Then, notice that irregular sampling is also useful in practice; we consider the general cases of the irregular sampling and establish a general irregular sampling theorem for multiwavelet subspaces. Finally, using this generalized irregular sampling theorem, we obtain an estimate for the perturbations of regular sampling in shift-invariant spaces.

1. Introduction

At the present time the sampling theorem plays a crucial role in signal processing and communication, as it establishes an equivalence between discrete signals and analogue (continuous) signals. For a band-limited signal, the classical Shannon sampling theorem provides an exact representation by its uniform samples with a sampling rate higher than its Nyquist rate. But there exist several problems. Firstly, real-world signals or images are never exactly band-limited. Secondly, there is no such device as an ideal (antialiasing or reconstruction) low-pass filter. Thirdly, Shannon reconstruction formula is rarely used in practice (especially with images) because of the slow decay of the sinc function. Therefore, this classical Shannon sampling theorem has been generalized to many other forms.

Extensions of Shannon sampling theorem to scalar wavelets can be found in [15], but a scalar wavelet cannot have the orthogonality, compact support, and symmetry at the same time (except the Haar wavelet). It is a disadvantage for signal processing. Meanwhile, multiwavelets have attracted much attention in the research community, since multiwavelet has more desired properties than any scalar wavelet function, such as orthogonality, short compact support, symmetry, high approximation order, and so on. The first orthogonal multiwavelet with symmetry, approximation order, and compact support was presented by Geronimo et al. [6]. In addition, the sampling theorems for multiwavelet subspaces were studied in [710]. The authors of [7, 9] presented the construction of compactly supported orthogonal multiscaling functions that are continuously differentiable and cardinal. The scaling functions thereby support a Shannon-like sampling theorem. However, the multiwavelets of [7, 9] do not have symmetry. It is not good for digital signal processing and image compression. They also did not study the sampling theorem for frame in multiwavelet subspaces, which is very important in application.

A reconstruction from more general sets of points is necessary if the measurements cannot be made at uniform points. Hence, irregular sampling is also useful in practice. In [11], the authors obtained a Feichtinger-Grochenig iterative algorithm based on the quasiinterpolation projection procedure to recover signals from irregular samples for multiwavelet subspaces. However, the maximal allowable gap between two sampling points needed for reconstructing a function from its samples was not obtained, which was supposed to exist in theory. The authors of [10] generalized the multiwavelet sampling theorem by reproducing a kernel and derived an estimate for the perturbations of uniform noninteger sampling in shift-invariant spaces, but their results just based on the Riesz basis.

In our paper, we will show a sufficient condition for regular sampling theorem to hold in multiwavelet subspaces for frames. Notice that a reconstruction from more general sets of points is necessary if the measurements cannot be made an uniform; we establish the irregular sampling theorems in multiwavelet subspaces. Finally, an estimate for the perturbations of regular sampling in shift-invariant spaces is derived.

This paper is organized as follows. Section 2 contains some definitions in this correspondence. Also, we review some relative notations. In Section 3, we discuss general uniform noninteger sampling and obtain a sufficient condition for uniform noninteger sampling theorem to hold. In Section 4, an irregular sampling theorem in general multiwavelet subspaces is established. Finally, by applying the result in Section 4, we estimate the perturbations of uniform noninteger sampling in shift-invariant spaces.

2. Preliminary

We now introduce some notations used in this correspondence.

is the space of continuous function.

The shift-invariant closed subspace generated by

For a function , we denote by the minimal closed shift invariant subspace that contains .

Let denote vector (we denote vectors and matrices in this paper in boldface). The integration is defined as

The Fourier transform of vector is defined by

The inverse Fourier transform of vector is written by

is the Zak transform of function . The Zak transform of vector is defined by

A collection of elements in a Hilbert space is called a frame if there exist constants and , , such that

If is a frame for , then there exists a dual frame for [12, Theorem ].

For ,we can write

Let The reproducing kernel is defined as where denotes the Hermitian conjugate. Put . Then where , , is constant, and for , .

Let , means the largest number in subset , and means the smallest number in subset .

For , let bracket function be the function defined a.e. by .

3. General Uniform Sampling Theorem

The main purpose of this section is to study the regular sampling theorem for frame in multiwavelet subspace.

Firstly, we start with some useful lemmas.

Lemma 3.1. Let . If and for all , then

Proof. Suppose that is a frame sequence in . Let , then there exists a sequence such that .
Define We have It is easy to check that and ; thus . Similar to the above argument, , we get .
Suppose that holds; by and , we can obtain It implies that The proof is completed.

Lemma 3.2. Let , then the following two assertions are equivalent:(a)for any , , converges pointwise to a continuous function;(b) and

Proof. (a)(b). It is easy to see that . For each , since is convergent for each , clearly, . For each , define Then is a bounded linear functional on with the norm . For any , define . Since is continuous on , we have By the Banach-Steinhaus theorem [13], that is, is bounded on .
(a)(b). By the Cauchy inequality, in convergent uniformly on , so the limit function is continuous.

Lemma 3.3. Let be a frame for with bounds A and B. If for all , , then is a frame for the subspace .

Proof. Suppose that is a frame for the subspace , then there exist constants such that Notice that for all , ; if , then . Hence, By the definition of Frame, is a frame for the closed subspace . Then we get the desired result.

Lemma 3.4. Let and , then there exists a set , such that for any and , converges pointwise to a continuous function.

Proof. By , then . Hence there exists a set such that . Notice that ; using Lemma 3.2, clearly, for any , converges pointwise to a continuous function in set .

From the above lemmas, we have the following result.

Theorem 3.5. Let be a frame for . Suppose that , are continuous functions, for all , , and Then holds for each and .

Proof. Suppose that is a frame for and , , then, by [14, Proposition ] and Lemma 3.3, there exist constants such that From Lemma in [15], we have the dual frame for , where is defined by
Hence, for all , holds.
So, it follows that
From above results, obviously, if then Hence, converges to in for any and .
By for all , , it is easy to check that is a dual frame for ; then we have for any and . So we get the desired result.

Based on these facts, the following sampling theorem is established.

Theorem 3.6. Let be a frame for . Suppose that for all , , and , are continuous such that If there exist constants and , such that where , then there exists a frame of , for , holds, where the convergence is in .

Proof. Define the function by , where is defined by (3.11). From Lemma 3.3, it is easy to see that and are the frames for the subspace .
Notice that From the proof process of Proposition in [14], is a frame for the subspace . Similar to the above argument, obviously, for , is the frame for the subspace . Hence, , there exist constants such that Thus, we have where is the dual frame for .
From above results, we get where .
Notice that , and are the frame for subspace ; by [16, Lemma ], we have (except a null measurable set).
By Lemma 3.3, it follows that for all , holds, where . Again by Lemma 3.3, we have , , then, From (3.23) and (3.25), , we get
For notice (3.26), we have
Let , then , there exist constants number such that Hence, according to the definition of frame, is the frame for the subspace . Let ; by Theorem 3.5, clearly, .
Notice that Using (3.27) and (3.22), then
For , by Lemmas 3.1 and 3.4, notice that for all , , then , for all , . Hence, holds. From (3.31) and (3.32), we get that is the frame for the subspace . Then there exists a dual frame of such that The proof is completed.

4. General Irregular Sampling Theorem

In many realistic situations, we try to reconstruct a continuous signal by using its discrete samples . Obviously, the samples cannot be arbitrary; we must discuss the values of deviation from the uniformly noninteger points. The following theorem studies this problem.

Based on the results in Section 3, we get the irregular sampling theorems.

Theorem 4.1. Let be a frame for with bounds A and B. Suppose that for all , , and , are continuous such that Moreover, suppose that there exist constants and , such that If there exists a sequence and two constants and , such that holds for any . Then, there exists a frame for such that for any , holds, where the convergence is both in and uniform on R.

Proof. From the argument of Theorem 3.6, it is easy to see that is a frame for the closed subspace of . Hence, there exist two constants and , such that
By (4.3), we have Then, is also a frame for the closed subspace of . So, there exists a dual frame of such that for any .

Theorem 4.2. Let be a frame for with bounds A and B. Suppose that for all , , and , are continuous such that Moreover, suppose that there exist constants and , such that If there exists a sequence and a constant such that holds for any , then, there exists a frame for such that for any holds, where the convergence is both in and uniform on R.

Proof. If then By Theorem 4.1, we get the desired result.

5. Perturbation of Uniform Noninteger Sampling in Shift-Invariant Spaces

Our objective is to find explicit formulas or algorithms to calculate the ranges of the perturbations of uniform noninteger sampling in shift-invariant spaces. In order to establish the algorithm for perturbations of uniform noninteger sampling for frame in shift-invariant spaces, we need to introduce the function class defined and used in [10].

Definition 5.1. and consists of all the measurable functions f, for which the norm

Theorem 5.2. Let be a frame for with bounds A and B. Suppose that there exist constants and , such that Moreover, suppose that for all , , and , are continuous such that Then, for any , , , there is a frame of such that holds in for any if

Proof. Notice that is a frame for , then, , there exists a square summable sequence such that holds in .
Similar to the argument of Theorem in [10], let Take Then, holds for any , , and Moreover Hence
From the argument in Theorem 3.6, is a frame for the closed subspace . By (3.25), then Again from the proof process in Theorem 3.6, obviously, holds, where is defined by then By Theorem 4.2, we only need to show that This is exactly implied by (5.5). We get the desired result.

6. Conclusion

Multiwavelets have orthogonality, regularity, short compact support, symmetry, and high approximation order. It is not possible in the scalar wavelet case. In our paper, we study the sampling theorem for frames in multiwavelet subspaces and obtain the sufficient condition under which the regular sampling theorem holds. Notice that the measurements may not be made at uniform points; we establish an irregular sampling theorem for frame in multiwavelet subspaces. Finally, an estimate for the perturbations of regular sampling in shift-invariant spaces is derived.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 10971228) and the Henan Provincial Natural Science Foundation of China (no. 102300410205).

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Copyright © 2010 Liu Zhanwei 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.


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