# A Short Note on Wavelet Frames Based on FMRA on Local Fields

Authors: M. Younus Bhat
Journal: Journal of Mathematics
 (2020)

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## Abstract

The concept of frame multiresolution analysis (FMRA) on local fields of positive characteristic was given by Shah in his paper, Frame Multiresolution Analysis on Local Fields published by Journal of Operators. The author has studied the concept of minimum-energy wavelet frames on these prime characteristic fields. We continued the studies based on frame multiresolution analysis and minimum-energy wavelet frames on local fields of positive characteristic. In this paper, we introduce the notion of the construction of minimum-energy wavelet frames based on FMRA on local fields of positive characteristic. We provide a constructive algorithm for the existence of the minimum-energy wavelet frame on the local field of positive characteristic. An explicit construction of the frames and bases is given. In the end, we exhibit an example to illustrate our algorithm.


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## Body

## 1. Introduction

LetK be a field and a topological space. Then, K is called a local field if both K+ and K∗ are locally compact abelian groups, where K+ and K∗ denote the additive and multiplicative groups of K, respectively. If K is any field and is endowed with the discrete topology, then K is a local field. Furthermore, if K is connected, then K is either ℝ or ℂ. If K is not connected, then it is totally disconnected. Hence, by a local field, we mean a field K which is locally compact, nondiscrete, and totally disconnected. The p-adic fields are examples of local fields. For more details, refer [1]. In the rest of this paper, we use the symbols ℕ,ℕ0 and ℤ to denote the sets of natural and nonnegative integers and integers, respectively.LetK be a local field. Let dx be the Haar measure on the locally compact abelian group K+. If α∈K and α≠0, then dαx is also a Haar measure. Let dαx=αdx. We call α the absolute value of α. Moreover, the map x⟶x has the following properties: (a) x=0 if and only if x=0; (b) xy=|x‖y| for all x,y∈K; and (c) x+y≤maxx,y for all x,y∈K. Property (c) is called the ultrametric inequality. The set D=x∈K:x≤1 is called the ring of integers in K. Define B=x∈K:x<1. The set B is called the prime ideal in K. The prime ideal in K is the unique maximal ideal in D, and therefore B is principal ideal as well as prime ideal. Since the local field K is totally disconnected, there exists an element of B of maximal absolute value. Let P be a fixed element of the maximum absolute value in B. Such an element is called a prime element of K. Therefore, for such ideal B in D, we have B=P=PD. As it was proved in [1], the set D is compact and open. Hence, B is compact and open. Therefore, the residue space D/B is isomorphic to a finite field GFq, where q=pk for some prime p and k∈ℕ.LetD∗=D∖B=x∈K:x=1. Then, it can be proved that D∗ is a group of units in K∗, and if x≠0, then we may write x=Pkx′,x′∈D∗. For a proof of this fact, refer [1]. Moreover, each Bk=PkD=x∈K:x<q−k is a compact subgroup of K+ and usually known as the fractional ideals of K+. Let U=aii=0q−1 be any fixed full set of coset representatives of B in D; then, every element x∈K can be expressed uniquely as x=∑ℓ=k∞cℓPℓ with cℓ∈U. Let χ be a fixed character on K+ that is trivial on D but is nontrivial on B−1. Therefore, χ is a constant on cosets of D, so if y∈Bk, then χyx=χyx,x∈K. Suppose that χu is any character on K+; then, clearly, the restriction χu|D is also a character on D. Therefore, if un:n∈ℕ0 is a complete list of the distinct coset representative of D in K+, then, as it was proved in [1], the set χun:n∈ℕ0 of distinct characters on D is a complete orthonormal system on D.The Fourier transformf^ of a function f∈L1K∩L2K is defined by(1)f^ξ=∫Kfxχξx¯dx.It is noted that(2)f^ξ=∫Kfxχξx¯dx=∫Kfxχ−ξxdx.Furthermore, the properties of the Fourier transform on local fieldK are much similar to those on the real line. In particular, Fourier transform is unitary on L2K.We now impose a natural order on the sequenceunn=0∞. We have D/B≅GFq, where GFq is a c-dimensional vector space over the field GFp. We choose a set 1=ζ0,ζ1,ζ2,…,ζc−1⊂D∗ such that span ζjj=0c−1≅GFq. For n∈ℕ0 satisfying(3)0≤n<q,n=a0+a1p+⋯+ac−1pc−1,0≤ak<p,k=0,1,…,c−1,we define(4)un=a0+a1ζ1+⋯+ac−1ζc−1P−1.Also, forn=b0+b1q+b2q2+⋯+bsqs,n∈ℕ0,0≤bk<q,k=0,1,2,…,s, we set(5)un=ub0+ub1P−1+⋯+ubsP−s.This definesun for all n∈ℕ0. In general, it is not true that um+n=um+un. However, if r,k∈ℕ0and0≤s<qk, then urqk+s=urP−k+us. Furthermore, it is also easy to verify that un=0 if and only if n=0 and uℓ+uk:k∈ℕ0=uk:k∈ℕ0 for a fixed ℓ∈ℕ0. Hereafter, we use the notation χn=χun,n≥0.Let the local fieldK be of characteristic t>0 and ζ0,ζ1,ζ2,…,ζc−1 be as above. We define a character χ on K as follows:(6)χζμP−j=exp2πij,μ=0andj=1,1,μ=1,…,c−1orj≠1.In 2015, Shah [2] introduced the concept of frame multiresolution analysis (FMRA) on local fields, which can be sought as an extension of multiresolution analysis (MRA) on local fields of positive characteristic. First of all, let us recall the definition of FMRA as given by Shah. Let K be a local field of positive characteristic p>0 and P be a prime element of K. A frame multiresolution analysis (FMRA) of L2K is a sequence of closed subspaces Vj:j∈ℤ of L2K satisfying the following properties:(a)
Vj⊂Vj+1,for allj∈ℤ(b)
∪j∈ℤVj is dense in L2K(c)
∩j∈ℤVj=0(d)
f⋅∈Vj if and only if fP−1⋅∈Vj+1,for allj∈ℤ(e)
There is a functionφ∈V0 such that φ⋅−uk:k∈ℕ0 forms a frame in V0The functionφ is called a frame refinable function. It is noted that the shifts of φ form a tight frame in the above FMRA. Replacing “a tight frame” in the above by “an orthonormal or a Riesz base” will arrive on the definition of a MRA on local fields of positive characteristic.A finite familyΨ:=ψ1,ψ2,…,ψL generates a wavelet frame for L2K if there exist positive numbers 0<A≤B<∞ such that, for all f∈L2K,(7)Af22≤∑ℓ=1L∑j∈ℤ∑k∈ℕ0f,ψj,kℓ2≤Bf22,where ψj,kℓ=qj/2ψℓP−j⋅−uk. The largest constant A and the smallest constant B satisfying the above are called the lower and upper wavelet frame bound, respectively. A wavelet frame is a tight wavelet frame if A and B are chosen so that A=B, and then the set Ψ:=ψ1,ψ2,…,ψL is called a set of generators for the corresponding tight wavelet frame. Furthermore, the wavelet frame is called a Parseval wavelet frame if A=B=1, i.e.,(8)∑ℓ=1L∑j∈ℤ∑k∈ℕ0f,ψj,kℓ2=f22,for allf∈L2K,and in this case, every function f∈L2K can be written as(9)fx=∑ℓ=1L∑j∈ℤ∑k∈ℕ0f,ψj,kℓψj,kℓx.A tight wavelet frameΨ is called a FMRA tight frame on local fields with frame bound 1 if Ψ⊂V1. Here, in this article, we are concerned with a minimum-energy wavelet frame which is more restrictive than a FMRA tight frame on local fields of positive characteristic. Here, we recall the definition of minimum-energy wavelet frames on local fields of positive characteristic [3].Definition 1.
Letφ∈L2K satisfy φ^∈L∞, φ^ be continuous at 0, and φ^0=1. Suppose that φ generates the nested closed subspaces Vj:j∈ℤ. Then, a finite family Ψ=ψ1,ψ2,…,ψL⊂V1 is called a minimum-energy wavelet frame associated with φ if(10)∑k∈ℕ0f,φ1,k2=∑k∈ℕ0f,φ0,k2+∑ℓ=1L∑k∈ℕ0f,ψ0,kℓ2,for allf∈L2K,where φj,k⋅=qj/2φP−1⋅−uk. By the Parseval identity, minimum-energy wavelet frame Ψ must be a tight frame for L2K with the frame bound being equal to 1. At the same time, the above equation is equivalent to(11)∑k∈ℕ0f,φ1,kφ1,k=∑k∈ℕ0f,φ0,kφ0,k+∑ℓ=1L∑k∈ℕ0f,ψ0,kℓψ0,kℓ,for allf∈L2K.
Motivated and inspired by various constructions of minimum-energy wavelet frames [4–10] and classical wavelet frames on finite fields [11–13], we, in this paper, discuss some constructions of minimum-energy wavelet frames which are based on the frame multiresolution analysis on local fields of positive characteristic. This paper is organized in the following manner. In Section 2, we present some preliminaries for the FMRA and the minimum-energy wavelet frames on local fields of positive characteristic. In Section 3, we present the main results. Here, we provide a constructive algorithm for the existence of the minimum-energy wavelet frame on the local field of positive characteristic. We also construct an example to illustrate our algorithm.


## 2. Notations and Preliminaries

Here, we present some preliminaries for the FMRA and the minimum-energy wavelet frames on local fields of positive characteristic.From the definition of FMRA on local fields, we know thatV0⊂V1=spanφP−1⋅−uk:k∈ℕ0¯. Since φ⋅∈V0, there exists a sequence hkk∈ℕ0∈l2ℕ0 such that(12)φx=q∑k∈ℕ0hkφP−1x−uk.The Fourier transform of (12) yields(13)φ^ξ=m0Pξφ^Pξ,where(14)m0ξ=1q∑k∈ℕ0hkχkξ,¯is an integral periodic function in L2D, where D=x∈K:x≤1 is the ring of integers in K and is often called the refinement symbol of φ. Observe that χk0=φ^0=1. Therefore, by letting ξ=0 in (13) and (14), we obtain ∑k∈ℕ0hk=1.ConsiderΨ=ψ1,ψ2,…,ψL⊂V1, with(15)ψℓx=q∑k∈ℕ0hkℓφP−1x−uk,ℓ=1,2,…,L.Equation (15) can be written in the frequency domain as(16)ψℓ^ξ=mℓPξφ^Pξ,where(17)mℓξ=1q∑k∈ℕ0hkℓχkξ¯,ℓ=1,2,…,L,are the integral periodic function in L2D and are called the framelet symbols or wavelet masks.Withmℓξ,ℓ=0,1,…,L, as framelet symbols, we formulate the q×L+1 matrix ℳξ as(18)ℳξ=m0ξ+Pu0m1ξ+Pu0⋯mLξ+Pu0m0ξ+Pu1m1ξ+Pu1⋯mLξ+Pu1⋮⋮⋱⋮m0ξ+Puq−1m1ξ+Puq−1⋯mLξ+Puq−1.Shah and Debnath [3] proved that if Ψ=ψ1,ψ2,…,ψL⊂V1 forms a minimum-energy wavelet frame in L2K, then the mask matrix ℳξ should satisfy certain conditions as follows.Lemma 1.
Suppose that the refinable functionφ and the framelet symbols mℓξ,ℓ=0,1,…,L, satisfy (13)–(17). If φ^ is continuous at 0 and φx generates a sequence of nested closed subspaces Vj:j∈ℤ, then the following statements are equivalent:(1)
Ψ is a minimum-energy frame associated with φ.(2)
ℳξℳ∗ξ=Iq,Iq is an identity matrix of order q (19).(3)
αr,s=∑k∈ℕ0hr−qkhs−qk+∑ℓ=1Lhr−qkℓhs−qkℓ−qδr,s=0,∀r,s∈ℕ0.
Lemma1 gives the necessary and sufficient condition for the existence of the minimum-energy wavelet frames associated with refinable function φ. However, it is not a good choice to use this theorem to construct the minimum-energy wavelet frames. For convenience, Shah and Debnath [3] presented some conditions in terms of the framelet symbols.Lemma 2.
Letφ∈L2K be the refinable function with refinement mask m0ξ such that φ^ is continuous at 0 and φ^0=1. If Ψ=ψ1,ψ2,…,ψL is the minimum-energy wavelet frame associated with φ, then(19)∑r=0q−1m0Pξ+Pur2≤1,for allξ∈K.
In this paper, we start with the orthogonal vectors to obtain an explicit construction for the minimum-energy wavelet frame on local fields of positive characteristic.


## 3. Main Results

From Lemma1, we know that if we want to obtain a minimum-energy wavelet frame on local fields of positive characteristic, we should find L functions whose masks satisfy (19). Note the correlation of the rows of ℳξ; we should remove this feature first. For this, we introduce the polyphase decomposition technique. Similar to [3], we write(20)m0ξ=1qχu0ξ¯f1ξ+χu1ξ¯f2ξ+⋯+χuq−1ξ¯fqξ,mℓξ=1qχu0ξ¯gℓ1ξ+χu1ξ¯gℓ2ξ+⋯+χuq−1ξ¯gℓqξ,where frξ and gℓr,r=1,2,…,q, are the polyphase decompositions of m0ξ and mℓξ, respectively, and all these functions are 1/q−periodic. Let(21)Pξ=f1ξg11ξ⋯gℓ1ξf2ξg12ξ⋯gℓ2ξ⋮⋮⋱⋮fq−1ξg1q−1ξ⋯gℓq−1ξ,Sξ=χu0χu1⋯χuq−1χu0χPu1+u1⋯χPu1+uq−1⋮⋮⋱⋮χu0χPuq−1+u1⋯χPuq−1+uq−1.Then, we can easily see thatℳξ=1/qSξPξ, and (19) is equivalent to(22)PξPξ∗=Iq.The difference of (19) and (25) is that the rows of (25) are irrelevant to one another. Since m0ξ+Pu02+m0ξ+Pu12=f1ξ+Pu02+f2ξ+Pu02, we have f1ξ+Pu02+f2ξ+Pu02≤1.Riesz lemma tells us that there can exist a functionf3ξ+Pu0 such that f1ξ+Pu02+f2ξ+Pu02+f3ξ+Pu02=1. When f1ξ+Pu02+f2ξ+Pu02=1, then f3ξ+Pu0=0, which is the special case of the orthonormal wavelet base on local fields of positive characteristic. With f3ξ+Pu0 in hand, we can have a vector, in fact, a unit column vector, ζ0=f1ξ+Pu0,f2ξ+Pu0,f3ξ+Pu0T. Now, we expect the existence of two more unit column vectors ζ1 and ζ2 such that(23)ζ0T⋅ζ1=ζ0T⋅ζ2=ζ1T⋅ζ2=0.In fact,ζ1 and ζ2 form an orthonormal fundamental system of solutions of the linear equation ζ0T⋅x=0. By straightforward calculation, an orthonormal fundamental system of solutions is(24)ζ1=−f3¯Δ,0,f1¯ΔT,ζ2=−f1f2¯Δ,Δ,−f2¯f3ΔT.Here,Δ2=f12+f32=1−f22. Here, we notice that if we choose Pξ as the first two rows of the following matrix(25)ζ0,ζ1,ζ2=f1−f3¯Δ−f1f2¯Δf20Δf3f1¯Δ−f2¯f3Δ,then Pξ is a unitary matrix which implies that the corresponding matrix ℳξ is a mask matrix. Moreover, the minimum-energy wavelet frame matrix ℳξ on local fields has the shape(26)Pξ=f1−f3Δ¯−f1f2¯Δf20ΔNξ,where Nξ is a square matrix which is also unitary.Given a refinement functionφx, the refinement mask m0ξ+Pu0 should satisfy m0ξ+Pu0+m0ξ+Pu1+m0ξ+Pu2=1, which is the same as to the first column of (26). So, we can select the orthonormal wavelet masks as those of (26). So, all the orthonormal wavelet masks are of the shape(27)ζ0,ζ1,ζ2=f1−f3Δ¯−f1f2¯Δf20Δf3f1Δ¯−f2¯f3Δ100Nξ.


## 4. Example

Let us consider the case of the Haar wavelet. The refinement function of the Haar wavelet on local fields will be given byφx=χDx, and its refinement mask will be m0ξ=1/q1+χξ¯. It can be observed that mξ satisfies m0ξ+Pu02+m0ξ+Pu12=1, and the corresponding polyphase decompositions are f1=f2=1/q. Here, we notice that the polyphase decompositions of the orthonormal wavelet masks mℓξ are gℓ1ξ=−1/q and gℓ2ξ=1/q. Hence, the wavelet masks are given by mℓξ=−1/q1−χℓξ¯.



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*Source: 3957064-2020-11-02.xml*