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In this paper, we discuss the B-spline wavelets introduced by Chui and Wang in [1]. The definition for B-spline wavelet packets is proposed along with the corresponding dual wavelet packets. The properties of B-spline wavelet packets are also investigated.

Spline wavelet is one of the most important wavelets in the wavelet family. In both applications and wavelet theory, the spline wavelets are especially interesting because of their simple structure. All spline wavelets are linear combination of B-splines. Thus, they inherit most of the properties of these basis functions. The simplest example of an orthonormal spline wavelet basis is the Haar basis. The orthonormal cardinal spline wavelets in

Wavelets are a fairly simple mathematical tool with a variety of possible applications. If

For a given multiresolution analysis and the corresponding orthonormal wavelet basis of

The outline of the paper is as follows. In

In this Section, we introduce B-spline wavelets (or simply B-wavelets) and some notions used in this paper.

Every mth order cardinal spline wavelet is a linear combination of the functions

The cardinal B-splines

We use the following convention for the Fourier transform,

The Fourier transform of the scaling function

For each

1)

2)

3)

4) for each

Following Mallat [

5)

6)

7)

These subspaces

where

Substituting the value of

This gives

So, (2) can be written as

which is called the two scale relation for cardinal B-splines of order

Chui and Wang [

with support

So, substituting (6) into (5), we have the two scale relation

where,

Let

with the corresponding two scale sequence

For the scaling function

such that

Now, we have

Taking the Fourier transform of (13), we have

where,

A necessary and sufficient condition for the duality relationship (12) is that

A proof of this statement is given in ( [

where,

We observe that

See ( [

If

We say that

and

Lemma 1 Let

Proof See ([

Theorem 1 Let

Proof Let us suppose that

Again by Lemma 1 and (22), we have

On the other hand, let (24) holds.

Now,

Also,

Thus,

Following Coifman and Meyer [

where

When

and for

We call

where,

So, (24) can be written as

Similarly, taking the Fourier transformation on both sides of (26), we have

where,

Using these conditions we can write

We are now in a position to investigate the properties of B-spline wavelet packets.

Theorem 2 Let

Proof Since

By using (27), (30) and (36), we obtain

Hence, by Lemma 1, (35) follows.

Theorem 3 Let

Proof By (27), (30) and (36), for

For the family of B-spline wavelet packets

generated by

where

may be written as

A generalization of the above result for other values of

Theorem 4 For the B-spline wavelet packets, the following two scale relation

holds for all

Proof In order to prove the two scale relation, we need the following identity, see ([

Taking the right-hand side of (38), and applying the identity (39), we have

Next, we discuss the duality properties between the wavelet packets

Lemma 2 For all

Proof We will prove (41) by induction on

where

according to the proof of Theorem 7.24 in [

Since

Thus, we have

This shows that (41) also holds for

Lemma 3 For all

Proof By applying the Fourier transform formulations of Equations (25) and (26) and using (42) and (34), we have as in the proof of Lemma 2 that