On Generalized Carleson Operators of Periodic Wavelet Packet Expansions

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.


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, 1 < < ∞, 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 ( , 1) summability method of the partial sum operator for Walsh-type wavelet packet expansion of functions from the block space, B , 1 < ≤ ∞, −1 + −1 =1 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 ( , 1) method and generalized Carleson operator have been obtained.
Each pair ( can be chosen as a pair of quadrature mirror filters associated with a multiresolution analysis, but this is not necessary.

Block Spaces. A dyadic -block is a function
∈ [0, 1) which is supported on some dyadic interval such that Let B denote the space of measurable functions on [0, 1) which has an expansion where each is a -block and the coefficients , ∈ Z satisfy The quasi norm of ∈ B is given as the infimum of ‖| ⋅ |‖ over all possible decompositions of into blocks Let ∈ B ; then using (12) and the fact that for each , ‖ ‖ ≤ | | 1/ −1 which implies that ‖ ‖ 1 ≤ 1; that is, B ⊂ 1 [0, 1). Moreover, for is a -block supported on = [0, 1) so [0, 1) ⊂ B .
The classical example to show that for each > 1 there exists ∈ B which belongs to none of the [0, 1)-space is the following. Let

Summation of Series by Arithmetic Means.
If a series 0 + 1 + 2 + ⋅ ⋅ ⋅ is not convergent, that is, if = 0 + 1 + 2 + ⋅ ⋅ ⋅ + 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 The Scientific World Journal 3 and the last term tends to zero if → 0. Consider is said to be summable to by Cesàro's means of order 1. We write But may tend to a limit even though does not, for example, the series Here the partial sums are alternately 1 and 0, and it is easily seen that → 1/2.

Generalized Carleson Operators. Let {̃} be a periodic
Walsh-type wavelet packet basis. For any function The Carleson operator G is defined by The generalized Carleson operator G is defined by The weak Carleson operator is defined by The generalized weak Carleson operator is define by The dyadic Carleson operator G is defined by The generalized dyadic Carleson operator G is define by It is easy to prove that G , and G are sublinear operators.
Walsh Functions and Their Properties.
Observe that the Walsh system is the family of wavelet packets obtained by considering = 0 , 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 ⊕ : The Scientific World Journal The operation ⊕ is defined for almost all , ∈ [0, 1]. With this definition, we have for every pair , for which ⊕ is defined, (Golubov et al. [6], page 11).

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.
where is the generalized weak Carleson operator defined by (26) and is a positive finite constant.
Theorem 5. If a function belongs to B -class, 1 < ≤ ∞, then where is the generalized weak Carleson operator.

Lemmas
For the proof of our theorems, the following lemmas are required.
where is an arbitrary constant.

Proof of Theorem 4
Fix > 0 and a -block supported on the dyadic interval ⊂ [0, 1); two cases are considered.

Proof of Theorem 5
be a function of B . Then due to the 1 convergence of the average sum defining . Since The Scientific World Journal This completes the proof of Theorem 5.

Applications
Following corollary can be deduced from our theorems.
This completes the proof of the corollary.