Wavelet Packet Frames on a Half Line Using the Walsh-Fourier Transform

In this paper, we study the construction of dyadic wavelet packet frames on a positive half line using the WalshFourier transform.  

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Introduction and Preliminaries
Frames in Hilbert space were introduced by Duffin and Schaffer [1] in 1952, in the context of non-harmonic Fourier series.Couple of years later, frames were brought to life by Daubechies, Grossmann and Meyer [2].Frames are generalizations of orthonormal basis.The linear independence property for a basis, which allows every vector to be uniquely represented as a linear combination.
The theory of frames are widely used in signal processing, data analysis, image compression and enhancement, coding theory, filtering of signals and many more.
Various authors studied the wavelet frames and dyadic wavelet frames such as Daubechies [9], Chui and Shi [10], Casazza and Christensen [11], Christensen [12,13], Protosov and Farkov [14], Farkov [15], Shah and Debnath [16], Ahmad and Iqbal [17,18].Motivated by these authors, in this paper, we extended our results to dyadic wavelet packet frames on the positive half line   .Let be the positive half line and .Let us denote the integer and fractional parts of a number by    x , respectively.Then, for each and any positive integer j, we set For each , these numbers are the digits of a binary expansion and there exists where , j j x y x are defined in (1).Moreover, we note that 0 x y y     , where  denotes the substitution modulo 2 on   .For The extension of the function 1  to is denoted by the equality the generalized Walsh functions    : where Note that the Walsh functions almost behave like characters with respect to dyadic addition, namely Thus, for each fixed y, equality (3) is valid for all x    except countably many of them.For , x y , w h e r e , , x y j j j j j x y x y x y x y and , j j x y are given by (1).Note that for all and .It is shown by Golubov et al. [19] that both the system The Walsh Fourier transform of a function is given by ( 4).The properties of the Walsh Fourier transform are quite similiar to those of the classical Fourier transform [19][20][21].In particular, , then By a dyadic interval of range n in , we mean intervals of the form Moreover, the dyadic topology is generated by the collection of dyadic intervals and each dyadic interval is both open and closed under the dyadic topology.Therefore, it follows that for each 0 j j   , the Walsh function


is piecewise constant and hence continuous.Thus for where is uniquely determined by the relationship x I x  By a Walsh polynomial, we mean a finite linear combination of Walsh functions.Thus, an arbitrary Walsh polynomial of order n can be written as where the j b are complex coefficients.Since x , for each , therefore, each Walsh polynomial is a dyadic step function and vice versa [19,21].
be the space of dyadic entire functions of order n, that is, the set of all functions which are con-stant on all intervals of range n.Thus, for every Clearly, each Walsh polynomial of order 1 2 n belongs to and each function in      is of compact support and so is its Walsh Fourier transform.Thus , we will consider the following set of functions: , The numbers A and B are called frame bounds.
 , the frame is said to be tight.The frame is called exact if it ceases to be a frame whenever any single element is deleted from the frame.
The continuous wavelet transformation of a L 2 -function f with respect to the wavelet  , which satisfies admissibility condition, is defined as: The term wavelet denotes a family of functions of the form Translation:

Wavelet Packets on  
We have the following sequence of functions due to Wickerhauser [22].For where The constants C and D are called frame bounds.If C = D, the frame is said to be tight.The frame is called exact if it ceases to be a frame whenever any single element is deleted from the frame.
where 0  is a scaling function and may be taken as a characteristic function.If we increase l, we get the following Since the set   defined by ( 9) is also dense in Therefore, the system given in ( 10) is frame for if the inequalities in (11) holds for all and so on.
Here l  's have a fixed scale but different frequencies.
They are Walsh functions in [0 .The functions l , for integers k, l with , form an orthonormal basis of . ,1) For every partition P of the non-negative integers into the sets of the form , the collection of functions

Dyadic Wavelet Packet Frames on  
For any function , we consider the system , , , .
By taking Walsh Fourier transform to (10), we obtain, Then, we call system (10) wavelet packet frame for m Now for each j   , let j F be the function defined by
, for all     and in view of (8), we have , Applying Parseval's formula and the fact that Furthermore, the iterated series in ( 14) is absolutely convergent.Proof.From (12) we have

Since
, and therefore, by the Levi's Lemma, we obtain Now we claim that the itrated series in ( 14) is absolutely convergent.To do this, let which means that for each Therefore, it suffices to prove that This fact shows that iterated series in ( 14) is absolutely convergent.

Main Results
with frame bound C and D, then and ), .
where where t I s  are mutually disjoint , therefore, for m   and ve j T   , we ha


By letting and consecutively, we which is the right inequality of (16). to prove the left inequality of ( 16), let In order , , , , .
I C I   .As we have already shown it that   ough to prove that as .By ( 12) and the Cauchy-Schwartz i lity, we given For 0 for each 1 2 j j T    .ying (18) and Appl (19) in (17), we obtain, Now by Chui and Shi [10] and Shah and Debnath [16], we get the desired results.
Proof.For a function , we have Applying the Cauchy-Schwarz inequality twice, we have Copyright © 2013 SciRes.AJCM Thus result follows.

R E T R A
C T E D obtained from a single function  by the operation of dilation and translation.define the following operators as follows: Hilbert space H is called a frame for H if there exists two +ve numbers A and B such that for any  ,