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The Scientific World Journal
Volume 2013 (2013), Article ID 379861, 10 pages
http://dx.doi.org/10.1155/2013/379861
Research Article

On Generalized Carleson Operators of Periodic Wavelet Packet Expansions

Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi 221005, India

Received 29 April 2013; Accepted 14 July 2013

Academic Editors: M. Mansour and E. Momoniat

Copyright © 2013 Shyam Lal and Manoj Kumar. 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

Three new theorems based on the generalized Carleson operators for the periodic Walsh-type wavelet packets have been established. An application of these theorems as convergence a.e. for the periodic Walsh-type wavelet packet expansion of block function with the help of summation by arithmetic means has been studied.

1. Introduction

Wavelet packet expansions have wide applications in engineering and technology. The Walsh-type wavelet packet expansions play an important role in signal processing, numerical analysis, and quantum mechanics. A family of nonstationary wavelet packets considered the smooth generalization of the Walsh functions having some of the same nice convergence properties for expansion of -function, , as the Walsh-Fourier series. Walsh-type wavelet packet expansion has been studied by the researchers Billard [1], Nielsen [2], Sjölin [3] and others. In 1966, at first, Carleson operator has been introduced by Lennart Carleson (Carleson [4]). Several important properties of this operator has been studied by researcher Nielsen [2]. In this paper, the pointwise convergence almost everywhere by arithmetic means or summability method of the partial sum operator for Walsh-type wavelet packet expansion of functions from the block space, has been studied. Generalized Carleson operators are introduced and some new properties of generalized Carleson operators are investigated. Specific convergence properties of Walsh-type wavelet packet expansions of block functions using method and generalized Carleson operator have been obtained.

2. Definitions and Preliminaries

Walsh-Type Wavelet Packets. To every multiresolution analysis for , an associated scaling function and a wavelet are given with the properties that is an orthonormal basis for .

We write

Let be the set of natural numbers. Let , be a family of bounded operators on of the form with a real-valued sequence in such that

Define the family of functions recursively by letting ,    and then for , where .

The family is basic non stationary wavelet packets. is an orthonormal basis for .

Moreover, is an orthonormal basis for .

Each pair can be chosen as a pair of quadrature mirror filters associated with a multiresolution analysis, but this is not necessary.

The trigonometric polynomials given by are called the symbols of the filters.

The Fourier transforms of (5) are given by

The Haar low-pass quadrature mirror filter is given by otherwise, and the associated high-pass filter is given by

Definition 1. Let be a family of non-stationary wavelet packets constructed by using a family of finite filters for which there is a constant, such that is the Haar filter for every . If is compactly supported then is called a family of Walsh-type wavelet packets.

Definition 2. Let be a family of Walsh-type basic wavelet packets. For , define the corresponding periodic Walsh-type wavelet packets by

From Fubini’s theorem, it follows that is an orthonormal basis for .

Block Spaces. A dyadic -block is a function which is supported on some dyadic interval such that , where . Let denote the space of measurable functions on which has an expansion where each is a -block and the coefficients satisfy

The quasi norm of is given as the infimum of over all possible decompositions of into blocks

Let ; then using (12) and the fact that for each which implies that ; that is, . Moreover, for is a -block supported on so .

The classical example to show that for each there exists which belongs to none of the -space is the following.

Let

Then , but for every .

Summation of Series by Arithmetic Means. If a series is not convergent, that is, if does not tend to a limit, it is some time possible to associate with the series a “sum” in a less direct way. The simplest such method is “summation by arithmetic means”. Let be the arithmetic mean of the partial sums of the given series.

If , then also ; for if , then and the last term tends to zero if . Consider

If as is said to be summable to by Cesàro’s means of order . We write

But may tend to a limit even though does not, for example, the series

Here the partial sums are alternately and , and it is easily seen that .

2.1. Generalized Carleson Operators

Let be a periodic Walsh-type wavelet packet basis. For any function , define

The Carleson operator is defined by

The generalized Carleson operator is defined by

The weak Carleson operator is defined by

The generalized weak Carleson operator is define by

The dyadic Carleson operator is defined by

The generalized dyadic Carleson operator is define by

It is easy to prove that and are sublinear operators.

Walsh Functions and Their Properties. The Walsh system is defined recursively on by letting

Observe that the Walsh system is the family of wavelet packets obtained by considering , and using the Haar filters in the definition of the nonstationary wavelet packets.

The Walsh system is closed under pointwise multiplication. Define the binary operator by where and . Then(see Schipp et al. [5]).

We can carry over the operator to the interval by identifying those with a unique expansion (almost all has such a unique expansion) by their associated binary sequence . For two such points , define

The operation is defined for almost all . With this definition, we have for every pair for which is defined, (Golubov et al. [6], page 11).

3. Main Results

In this paper, three new theorems for the generalized Carleson operators on the periodic Walsh-type wavelet packets have been determined in the following form.

Theorem 3. Let be a periodic Walsh-type wavelet packet basis. Then for every -block , where is the generalized dyadic Carleson operator defined by (28) and is a positive finite constant.

Theorem 4. Let be a periodic Walsh-type wavelet packet basis. Then for every -block , where is the generalized weak Carleson operator defined by (26) and is a positive finite constant.

Theorem 5. If a function belongs to -class, , then where is the generalized weak Carleson operator.

4. Lemmas

For the proof of our theorems, the following lemmas are required.

Lemma 6 (Nielsen [7]). Let , and define recursively by
Then where .

Lemma 7 (Zygmund [8], page 3). Consider where for ; it is also called Abel’s transformation.

Lemma 8. Let be the Walsh system. Then where is a finite positive constant, and for all pairs for which is defined.

Proof. The Dirichlet kernel, , for the Walsh system satisfies (see Golubov et al. [6], page 21).
Hence, where (32), (34), and the fact that is a constant on dyadic intervals of the form are used. This completes the proof of Lemma 8.

Lemma 9. If then where is an arbitrary constant.

Proof. The kernel can be expanded as
Therefore, using Lemma 8, where indicates that only the terms for which and , respectively, should be included in the sum. This implies the estimate since . This completes the proof of Lemma 9.

5. Proof of Theorem 3

The dyadic arithmetic mean of partial sums for the expansion of a measurable (integrable) function in the periodic Walsh-type wavelet packets, holds everywhere with the arithmetic mean of the projection onto the (periodized) scaling space associated with the underlying multiresolution analysis (Hess-Nielsen and Wickerhauser [9]). Therefore, it suffices to consider the arithmetic mean of the projection operators on to the space .

Suppose that the -block is associated with the dyadic interval . If , then , and using the fact that the operator (and thus ) is of strong type . We have

Now suppose that with . Put , and define . We have

Fix , and let denote the operator kernel associated with the projection operators . Then there exists a finite constant (independent of ) such that (see Terence [10]).

Using the estimate (52) on the kernel , we obtain

Since and implies that , therefore, Finally we obtain where is independent of and and hence Theorem 3 follows.

6. Proof of Theorem 4

Fix and a -block supported on the dyadic interval ; two cases are considered.

Case I. If , then . Therefore, using Theorem 5.1. [7], page , we have

Case II. Let with . Let and define . Then
Notice that with
For , we have
Hence, it suffices to estimate with
Fix ; then which implies that whenever , there is an increasing sequence for which for some fixed and for . Since , therefore
Using that and the same technique as in the proof of Lemma 9, we complete the proof to conclude that and consequently which completes the proof of Theorem 4.

7. Proof of Theorem 5

Let be a function of . Then due to the convergence of the average sum defining . Since

therefore

This completes the proof of Theorem 5.

8. Applications

Following corollary can be deduced from our theorems.

Corollary 10. Let be a periodic Walsh-type wavelet packet basis. Then the Fourier expansion of any function , in is summable by arithmetic means pointwise a.e.

Proof. Let us write and
With , let , and observe that . For each , write
Thus
From this it follows that
This completes the proof of the corollary.

Acknowledgments

Shyam Lal, one of the authors, is thankful to DST-CIMS for encouragement to this work. Manoj Kumar is grateful to CSIR, India in the form of Junior Research Fellowship vide Reference no. 17-06/2012 (i)EU-V dated 28-09-2012 for this research work.

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