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.

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

We write

Let

Define the family of functions

The family

Moreover,

Each pair

The trigonometric polynomials given by

The Fourier transforms of (

The Haar low-pass quadrature mirror filter

Let

Let

From Fubini’s theorem, it follows that

The quasi norm of

Let

The classical example to show that for each

Let

Then

If

If

But

Here the partial sums

Let

The Carleson operator

The generalized Carleson operator

The weak Carleson operator

The generalized weak Carleson operator

The dyadic Carleson operator

The generalized dyadic Carleson operator

It is easy to prove that

Observe that the Walsh system is the family of wavelet packets obtained by considering

The Walsh system is closed under pointwise multiplication. Define the binary operator

We can carry over the operator

The operation

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.

Let

Let

If a function

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

Let

Then

Consider

Let

The Dirichlet kernel,

Hence,

If

The kernel can be expanded as

Therefore, using Lemma

The dyadic arithmetic mean of partial sums for the expansion of a measurable (integrable) function

Suppose that the

Now suppose that

Fix

Using the estimate (

Since

Fix

If

Let

Notice that

For

Hence, it suffices to estimate

Fix

Using that

Let

therefore

This completes the proof of Theorem

Following corollary can be deduced from our theorems.

Let

Let us write

With

Thus

From this it follows that

This completes the proof of the corollary.

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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