ON THE CHARACTERIZATION OF SCALING FUNCTIONS ON NON-ARCHEMEDEAN FIELDS

In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. This gap was filled by Gabardo and Nashed [11] by establishing a constructive algorithm based on the theory of spectral pairs for constructing non-uniform wavelet basis in L(R). In this setting, the associated translation set Λ = {0, r/N} + 2Z is no longer a discrete subgroup of R but a spectrum associated with a certain onedimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we characterize the scaling function for non-uniform multiresolution analysis on local fields of positive characteristic (LFPC). Some properties of wavelet scaling function associated with non-uniform multiresolution analysis (NUMRA) on LFPC are also established.


Introduction
Multiresolution analysis (MRA) is an important mathematical tool since it provides a natural framework for understanding and constructing discrete wavelet systems. The concept of MRA provides a natural framework for understanding and constructing discrete wavelet systems. Multiresolution analysis is an increasing family of closed spaces {V j : j ∈ Z} of L 2 (R) such that j∈Z V j = {0} and j∈Z V j is dense in L 2 (R) which satisfies f ∈ V j if and only if f (2·) ∈ V j+1 . Moreover, there exists a function ϕ ∈ V 0 such that the collection of integer translates of the function ϕ, {ϕ(· − k) : k ∈ Z}, represents a complete orthonormal system for V 0 . The function ϕ is called scaling function or father wavelet. The concept of multiresolution analysis has been extended in various ways in recent years. These concepts are generalized to L 2 R d , to lattices different from Z d , allowing the subspaces of MRA to be generated by Riesz basis instead of orthonormal basis, admitting a finite number of scaling functions, replacing the dilation factor 2 by an integer M ≥ 2 or by an expansive matrix A ∈ GL d (R) as long as A ⊂ AZ d . All these concepts are developed on regular lattices, that is the translation set is always a group. Recently, Gabardo and Nashed [11] considered a generalization of Mallat's [21] celebrated theory of MRA based on spectral pairs, in which the translation set acting on the scaling function associated with the MRA to generate the subspace V 0 is no longer a group, but is the union of Z and a translate of Z. Based on one-dimensional spectral pairs, Gabardo and Yu [12] considered sets of nonuniform wavelets in L 2 (R). In the heart of any MRA, there lies the concept of scaling functions. Cifuentes et al. [10] characterized the scaling function of MRA in a general settings. The multiresolution analysis whose scaling functions are characteristic functions some elementary properties of MRA of L 2 (R n ) are established by Madych [20]. Zhang [26] studied scaling functions of standard MRA and wavelets. Zhang [26] characterized support of the Fourier transform of scaling functions.
The theory of wavelets, wavelet frames, multiresolution analysis, Gabor frames on local fields of positive characteristics (LFPC) are extensively studied by many researchers including Benedetto, Behera and Jahan, Ahmed and Neyaz, Ahmad and Shah, Jiang, Li and Ji in the references [1-4, 7-9, 13, 19, 22, 24] but still more concepts required to be studied for its enhancement on LFPC. Albeverio, Kozyrev, Khrennikov, Shelkovich, Skopina and their collaborators also established the theory of MRA and wavelets on the p-adic field Q p in a series of papers [5,6,[14][15][16][17][18], where Q p is a local field of characteristic 0. Recently, Shah and Abdullah [23] have generalized the concept of multiresolution analysis on Euclidean spaces R n to nonuniform multiresolution analysis on local fields of positive characteristic, in which the translation set acting on the scaling function associated with the multiresolution analysis to generate the subspace V 0 is no longer a group, but is the union of Z and a translate of Z, where Z = {u(n) : n ∈ N 0 } is a complete list of (distinct) coset representation of the unit disc D in the locally compact Abelian group K + . More precisely, this set is of the form Λ = {0, r/N } + Z, where N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime. They call this a nonuniform multiresolution analysis on local fields of positive characteristic. Inspired by the work of Shah and Abdullah [23], we in this paper establish the characterization of scaling function for nonuniform multiresolution on local fields of positive characteristic. Some properties of wavelet scaling functions associated with NUMRA on LFPC are established.
The remainder of the paper is structured as follows. In Section 2, we discuss preliminary results on local fields as well as some definitions and auxiliary results. Section 3 is devoted to the characterization of scaling function associated with nonuniform multiresolution analysis on LFPC.

Local fields
A local field K is a locally compact, non-discrete and totally disconnected field. If it is of characteristic zero, then it is a field of p-adic numbers Q p or its finite extension. If K is of positive characteristic, then K is a field of formal Laurent series over a finite field GF (p c ). If c = 1, it is a p-series field, while for c = 1, it is an algebraic extension of degree c of a p-series field. Let K be a fixed local field with the ring of integers Since K + is a locally compact Abelian group, we choose a Haar measure dx for K + . The field K is locally compact, non-trivial, totally disconnected and complete topological field endowed with non-Archimedean norm | · | : K → R + satisfying (c) |x + y| ≤ max {|x|, |y|} for all x, y ∈ K.
Property (c) is called the ultrametric inequality. Let B = {x ∈ K : |x| < 1} be the prime ideal of the ring of integers D in K. Then, the residue space D/B is isomorphic to a finite field GF (q), where q = p c for some prime p and c ∈ N. Since K is totally disconnected and B is both prime and principal ideal, so there exist a prime element p of K such that B = p = pD. Let Clearly, D * is a group of units in K * and if x = 0, then can write x = p n y, y ∈ D * . Moreover, if U = {a m : m = 0, 1, . . . , q − 1} denotes the fixed full set of coset representatives of B in D, then every element x ∈ K can be expressed uniquely as Recall that B is compact and open, so each fractional ideal is also compact and open and is a subgroup of K + . We use the notation in Taibleson's book [25].
In the rest of this paper, we use the symbols N, N 0 and Z to denote the sets of natural, non-negative integers and integers, respectively. Let χ be a fixed character on K + that is trivial on D but non-trivial on B −1 . Therefore, χ is constant on cosets of D so if y ∈ B k , then χ y (x) = χ(y, x), x ∈ K. Suppose that χ u is any character on K + , then the restriction χ u |D is a character on D. Moreover, as characters on D, then, as it was proved in [25], the set χ u(n) : n ∈ N 0 of distinct characters on D is a complete orthonormal system on D.
We now impose a natural order on the sequence Also, for Further, it is also easy to verify that u(n) = 0 if and only if n = 0 and for a fixed ℓ ∈ N 0 . Hereafter we use the notation χ n = χ u(n) , n ≥ 0. Let the local field K be of characteristic p > 0 and ζ 0 , ζ 1 , ζ 2 , . . . , ζ c−1 be as above. We define a character χ on K as follows: exp(2πi/p), µ = 0 and j = 1, 1, µ = 1, . . . , c − 1 or j = 1.

Fourier transforms on local fields
The Fourier transform of f ∈ L 1 (K) is denoted byf (ξ) and defined by It is noted thatf The properties of Fourier transforms on local field K are much similar to those of on the classical field R. In fact, the Fourier transform on local fields of positive characteristic have the following properties: • The map f →f is a bounded linear transformation of is called the Fourier series of f . From the standard L 2 -theory for compact Abelian groups, we conclude that the Fourier series of f converges to f in L 2 (D) and Parseval's identity holds:

Nonuniform MRA on local fields
Definition 1. For an integer N ≥ 1 and an odd integer r with 1 ≤ r ≤ qN − 1 such that r and N are relatively prime, we define It is easy to verify that Λ is not a group on local field K, but is the union of Z and a translate of Z.
Following is the definition of nonuniform multiresolution analysis (NUMRA) on local fields of positive characteristic given by Shah and Abdullah [23].

Definition 2.
For an integer N ≥ 1 and an odd integer r with 1 ≤ r ≤ qN − 1 such that r and N are relatively prime, an associated NUMRA on local field K of positive characteristic is a sequence of closed subspaces {V j : j ∈ Z} of L 2 (K) such that the following properties hold: (e) There exists a function ϕ in V 0 such that {ϕ(· − λ) : λ ∈ Λ}, is a complete orthonormal basis for V 0 .
It is worth noticing that, when N = 1, one recovers the definition of an MRA on local fields of positive characteristic p > 0. When, N > 1, the dilation is induced by p −1 N and |p −1 | = q ensures that qN Λ ⊂ Z ⊂ Λ. For every j ∈ Z, define W j to be the orthogonal complement of V j in V j+1 .
Then we have It follows that for j > J, where all these subspaces are orthogonal. By virtue of condition (b) in the Definition 2, this implies a decomposition of L 2 (K) into mutually orthogonal subspaces. As in the standard scheme, one expects the existence of qN −1 number of functions so that their translation by elements of Λ and dilations by the integral powers of p −1 N form an orthonormal basis for L 2 (K).
Let a and b be any two fixed elements in K. Then, for any prime p and m, n ∈ N 0 , let D p , T u(n)a and E u(m)b be the unitary operators acting on f ∈ L 2 (K) defined by: , , .
Lemma 2. Let (V j , ϕ) be non-uniform multiresolution analysis, where Then the necessary and sufficient condition for the existence of associated wavelets is

Characterization of scaling functions on LFPC
In this section, we establish the characterization of scaling functions associated with nonuniform multiresolution analysis on LFPC. We also provide the sufficient condition for the frequency band of the scaling function on LFPC.
Next we show that (ii) holds. Let f ∈ L 2 (K) be such that f (γ) = Φ q 2 D (γ). Then As (V j , ϕ) is NUMRA so if P j is orthogonal projection onto V j , we must have That is P j f → f as j → ∞.
Indeed for any fixed j ∈ Z by using (4.1), we have

This gives
Now invoking the Lesbesgue-dominated convergence theorem, we obtain Thus That is we get h(ξ) = 1 a.e. ξ ∈ q 2 D. Hence (ii) is proved.
Conversely, let ϕ ∈ L 2 (K) satisfying (i)-(iii). We define closed subspaces V j of L 2 (K) in the following way. For We will show (V j , ϕ) forms wavelet NUMRA. Using Lemma 1, the sequence {T λ ϕ} λ∈Λ is an orthonormal basis for V 0 .
By definition of V j , it can be easily shown that f (γ) ∈ V j if and only if where m 1 j , m 2 j are locally integrable, q-periodic functions. Let f ∈ V j , then as {T λ ϕ} λ∈Λ is an orthonormal basis for V 0 , so there exist {c j λ } ∈ ℓ 2 (N 0 ) such that On taking Fourier transform of both sides, we obtain where m 1 j and m 2 j are locally integrable and q-periodic functions. If f ∈ L 2 (K) satisfies for some m 1 j and m 2 j are locally integrable and q-periodic functions, then we can write for some scalars {c j k } and {d j k } k∈N 0 ∈ ℓ 2 (N 0 ). Therefore for some {l j λ } λ∈Λ ∈ ℓ 2 (N 0 ). By taking inverse Fourier transform on both sides, we obtain Hence V j (j ∈ Z) are given by (4.4). Now we are ready to show that V 0 ⊆ V 1 . Let f (γ) ∈ V 0 . Then by (4.4), we can write where m 1 0 and m 2 0 are locally integrable, q-periodic functions. Therefore, where This gives Using the conditions (i) and (iii), it can be easily shown that functions m 1 (γ) and m 2 (γ) are bounded. Also since m 1 (γ), m 2 (γ) and G(γ) are q-periodic, therefore the functions G(γ)m 1 (γ) and G(γ)m 2 (γ) are q-periodic and Thus by using (4.4)-(4.6), we infer that f (γ) ∈ V 1 . Hence V 0 ⊆ V 1 .
To prove that j∈Z V 0 = L 2 (K), it sufficient to show that, for any f ∈ L 2 (K), we have where P j is the orthonormal projection onto V j . Let f ∈ L 2 (K) be such that f ∈ C c (K). Now we have Since f has compact support, we can choose j so large that Then, using the fact that { √ qχ u(k) (ξ)} is an orthonormal basis for L 2 (q 2 D) and by (4.7), we get Putting (p −1 N ) j ξ = η in (4.8) and invoking the Lesbesgue-dominated convergence theorem, we get Thus the proof is complete.
In the context of Fourier domain, the following theorem gives necessary condition for scaling function of wavelet NUMRA on LFPC. as | ϕ| is continuous. By virtue of Lebesgue dominated convergence theorem, we obtain | ϕ(0)| = 1.
The following theorem gives the sufficient conditions for the frequency band of the scaling function of wavelet NUMRA on LFPC.

Conclusion
In the present paper, we have given a complete characterization of the scaling function for the non-uniform multiresolution analysis on local fields of positive characteristic. Theorem 1 characterizes the nonzero square integrable functions on L 2 (K) to be a scaling functions for the wavelet NUMRA by means of three simple conditions. Furthermore Theorem 3 expresses a compact subset of K to be the band scaling function of wavelet NUMRA on LFPC by means of three conditions. The present study can be extended in fractional settings and in the context of Multiresolution Analysis associated with Linear Canonical Transform.