Wavelet packets and wavelet frame packets on local fields of positive characteristic

Abstract

Using a prime element of a local field K of positive characteristic p, the concepts of multiresolution analysis (MRA) and wavelet can be generalized to such a field. We prove a version of the splitting lemma for this setup and using this lemma we have constructed the wavelet packets associated with such MRAs. We show that these wavelet packets generate an orthonormal basis by translations only.We also prove an analogue of splitting lemma for frames and construct the wavelet frame packets in this setting.[15] gave a definition of MRA on a local field of positive characteristic p and, similar to R n , have constructed the wavelets from an MRA.In this article we construct the wavelet packets associated with such an MRA. We also generalize the wavelet frame packets to this setup. First of all, we will discuss about wavelet packets very briefly.Let {V j : j ∈ Z} be an MRA of L 2 (R) with scaling function ϕ and wavelet ψ.Let W j be the corresponding wavelet subspaces: W j = span{ψ jk : k ∈ Z}. In the construction of a wavelet from an MRA, essentially the space V 1 is split into two orthogonal components V 0 and W 0 . Note that V 1 is the closure of the linear span of the functions {2 1/2 ϕ(2·−k) : k ∈ Z}, whereas V 0 and W 0 are respectively the closure of the span of {ϕ(·−k) : k ∈ Z} and {ψ(·−k) : k ∈ Z}. Since ϕ(2·−k) = ϕ (2(· − 2 −1 k)), we see that the above procedure splits the half integer translates of a function into integer translates of two functions.In a similar way, we can split W j , which is the span of {ψ(2 j · −k) : k ∈ Z} = {ψ (2 j (· − 2 −j k)) : k ∈ Z}, to get two functions whose 2 −(j−1) k translates will span the same space W j . Repeating the splitting procedure j times, we get 2 j functions whose integer translates alone span the space W j . If we apply this to each W j , then the resulting basis of L 2 (R) will consist of integer translates of a countable number of functions (instead of all dilations and translations of the wavelet ψ). This basis is called the "wavelet packet basis". The concept of wavelet packets was introduced by Coifman, Meyer and Wickerhauser [9,10]. For a nice exposition of wavelet packets of L 2 (R) with dilation 2, we refer to [14].The concept of wavelet packet was subsequently generalized to R n by taking tensor products [8]. The non-tensor product versions are due to Shen [25] for dyadic dilation, and Behera [2] for MRAs with a general dilation matrix and several scaling functions.Other notable generalizations are the biorthogonal wavelet packets [7], non-orthogonal

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“…We will say that the element a defines the elements of the array A (2) , and so on. The whole sum specified in (12) 0,0,...,0 = λ 0,0,...,0 λ 0,0,...,0 .…”

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