Some Results on Wavelet Frame Packets

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.


Introduction
Let us consider an orthonormal wavelet of ( ) For some applications, it is more convenient to have orthonormal bases with better frequency localization.This will be provided by the wavelet packets.
The wavelet packets introduced by Coifman, Meyer and Wickerhauser [1] [2] played an important role in the applications of wavelet analysis.But the theory itself is worthy for further study.Some developments in the wavelet packet theory should be mentioned, for instance Shen [3] generalized the notion of univariate orthogonal wavelet packets to the case of multivariate wavelet packets.Chui and Li [4] generalized the concept of orthogonal wavelet packets to the case of nonorthogonal wavelet packets.Yang [5] constructed a-scale orthogonal multiwavelet packets which were more flexible in applications.In [6], Chen and Cheng studied compactly supported orthogonal vector-valued wavelets and wavelet packets.Other notable generalizations are biorthogonal wavelet packets [7] and non-orthogonal wavelet packets with r-scaling functions [8].For a nice exposition of wavelet packets of ( ) 2 L  , see [9].
The main tool used in the construction of wavelet packets is the splitting trick [10].Let { } : j V j∈  be an MRA of ( ) 2 L  with the corresponding scaling function φ and the wavelet ψ .Let j W be the correspond- ing wavelet subspaces { } , = span : In the construction of a wavelet from an MRA, the space 1 V is split into two orthogonal components 0 V and 0 W , where 1 V is the closure of the linear span of the func- tions ∈ and 0 V and 0 W are the closure of the span of ( ) , we see that the above procedure splits the half integer translates of a function into the integer translates of two functions.
We can also choose to split j W which is the span of We then have two functions whose translates will span the same space j W . Repeating the splitting pro- cedure j times, we get 2 j functions whose integer translates alone span the space j W .If we apply this to each j W , then the resulting basis of ( ) 2 L  will give us a better frequency localization.This basis is called "wavelet packet basis".
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  was studied in [11], and the frame packets on d  were studied by Long and Chen in [12] [13].Also, multi- wavelet packets and frame packets of ( ) were discussed in [14].Throughout the paper, the space of all square integrable functions on the real line will be denoted by , f f f = and the relationship between functions and their Fourier transform is defined by ˆ2π , , f g f g = . For ( ) ( ) L ∞  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 ( )

Wavelet Packets and Wavelet Frame Packets
Definition 1.A multiresolution analysis (MRA) consists of a sequence of closed subspaces j V , j ∈  of ( )

2
L  and a function 0 V φ ∈ , such that the following conditions hold: T φ ∈ is an orthonormal basis for 0 V .

The function φ is called the scaling function of the given MRA.
Suppose that φ generates a multiresolution analysis and that there exists some function ψ in ( ) If ψ is a basic wavelet relative to φ , then it is clear that the wavelet spaces j W generated by ψ , satisfy the following properties: for all x ∈  .For the Haar basis, we have Therefore, for the Haar basis, the scaling function and the wavelet function satisfy the following recurrence equation ( ) ( ) ( ) Due to Coifman, Meyer and Wickerhauser [1] [2], we have the following sequences of functions is the filter which satisfies the following properties 2 2 , , 2, 7) and ( 8), we get ( ) ( ) ( ) ( ) ( ) ( ) and so on.
 is called a wavelet basis packet, where n is the oscillation parameter, j the scaling parameter and k the translation parameter.
We can also write ( ) . The family { } ; , n j k


constitutes wavelet frame packets if there are constants  and  , 0
 be the basic wavelet packets such that ( ) ( ) .

Then { }
; , n j k  constitutes wavelet frame packets with frame bounds  and  .Proof.Let  be the class of all those functions ( ) and f is compactly supported in \ 0  .By using the Parseval identity, we have ( ) ( ) Let ( ) ( ) ( ) which is 2π -periodic and whose Fourier coefficients are ( ) F k , k ∈  , then by Poisson sum formula we have, ( ) ( ) But the left side of ( 13) equals It follows that Applying (15) when where, In the expression for I , the parameter k is a non-zero integer.For each such k there is a unique non- negative integer l and a unique odd integer q such that 2 l k q = . Therefore, we have for all f ∈  .By using Schwarz's inequality we have  By changing variables in the second integral and using the fact that ( ) ( ) , and applying Schwarz's inequality for series we have ( ) These inequalities together with (16) give us , Since  is dense in ( )

2
L  , the above inequality holds for all ( ) Theorem 2. The system { } .
a.e.The converse is immediate.
Performing a change of variables, we see that ; , ; , ;0, ;0, , , ; this tells us that the system Replacing k by 2l , we have By Cauchy-Schwarz inequality, we get ( ) On solving the second term in the last product, we have

2 L
 .The orthonormal wavelet bases { } a frequency localization which is proportional to 2 j at the resolution level j .If we consider a bandlimited wavelet ψ (i.e.ψ is compactly supported), the measure of supp ( ) wavelet bases have poor frequency localization when j is large.

2 L  will be denoted by 1 2
corresponds to our scaling function φ and ( ) 1 x  corresponds to the wavelet ψ .If we increase n , we get the following structures

Theorem 3 .=
Let { } n  be a sequence of wavelet frame packets with bounds  and  .Define { } defined by (20) is a wavelet frame packet with bounds k and l .
can prove the upper frame condition.