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The aim of this paper is to study wavelet frame packets in which there are many frames. It is a generalization of wavelet packets. We derive few results on wavelet frame packets and have obtained the corresponding frame bounds.

Let us consider an orthonormal wavelet of

where

The wavelet packets introduced by Coifman, Meyer and Wickerhauser [

The main tool used in the construction of wavelet packets is the splitting trick [

tions

We can also choose to split

then have two functions whose

There are many orthonormal bases in the wavelet packets. Efficient algorithms for finding the best possible basis do exist; however for certain wavelet applications in signal analysis, frames are more suitable than orthonormal bases, due to the redundancy in frames. Therefore, it is worthwhile to generalize the construction of wavelet packets to wavelet frame packets in which there are many frames. The wavelet frame packets on

Throughout the paper, the space of all square integrable functions on the real line will be denoted by

and

respectively. Also the norm of any

tween functions and their Fourier transform is defined by

Fourier transform

be the collection of almost everywhere (a.e.) bounded functions, i.e., functions bounded everywhere except on sets of (Lebesgue) measure zero and equipped with the norm

Definition 1. A multiresolution analysis (MRA) consists of a sequence of closed subspaces

1)

2)

3)

4)

5)

The function

Suppose that

If

6)

7)

8)

Since both the scaling function

for all

Therefore, for the Haar basis, the scaling function and the wavelet function satisfy the following recurrence equation

Due to Coifman, Meyer and Wickerhauser [

where

where

and

For

and so on. The functions

Definition 2. The family

We can also write

Define

Consider

and

Theorem 1. Let

and

Then

Proof. Let

Since,

Hence,

Let

which is

Hence,

But the left side of (13) equals

It follows that

Applying (15) when

where,

In the expression for

Thus,

for all

By changing variables in the second integral and using the fact that

Hence,

These inequalities together with (16) give us

Since

Theorem 2. The system

and

Proof. By using the Plancherel theorem we have

Thus,

Lemma 1. If

for all

Proof. Let

Replacing

This shows that

By using (17) and (18), we have

Theorem 3. Let

If the numbers

then

Proof. Let

By Cauchy-Schwarz inequality, we get

On solving the second term in the last product, we have

Thus,

By (23), we have

Thus,

Similarly, one can prove the upper frame condition.