URAL MATHEMATICAL JOURNAL, Vol. 7, No. 1, 2021, pp. 3-15

DOI: 10.15826/umj.2021.1.001

ON THE CHARACTERIZATION OF SCALING FUNCTIONS ON NON-ARCHEMEDEAN FIELDS

Ishtaq Ahmed^, Owias Ahmad^, Neyaz Ahmad Sheikh"^

National Institute of Technology, Jammu and Kashmir, Srinagar-190006, India

tishtiyaqahmadun@gmail.com, ttsiawoahmad@gmail.com, tttneyaznit@yahoo.co.in

Abstract: 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 L2(R). In this setting, the associated translation set A = {0,r/N} + 2 Z is no longer a discrete subgroup of R but a spectrum associated with a certain one-dimensional 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.

Keywords: Scaling function, Fourier transform, Local field, NUMRA

1. 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 {Vj : j € Z} of L2(R) such that HjeZ Vj' = {0} and (JjeZ Vj- is dense in L2(R) which satisfies f € Vj if and only if f (2-) € Vj-+1. Moreover, there exists a function f € V0 such that the collection of integer translates of the function <p, {f (■ — k) : k € Z}, represents a complete orthonormal system for V0. The function f 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 L2(Rd), to lattices different from Zd, 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 € GLd(R) as long as A c AZd. 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 V0 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 L2(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 L2(Rn) 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 Qp in a series of papers [5, 6, 14-18], where Qp is a local field of characteristic 0. Recently, Shah and Abdullah [23] have generalized the concept of multiresolution analysis on Euclidean spaces Rn 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 V0 is no longer a group, but is the union of Z and a translate of Z, where Z = {u(n) : n € No} 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 A = {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.

2. Preliminaries on local fields

2.1. 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 Qp or its finite extension. If K is of positive characteristic, then K is a field of formal Laurent series over a finite field GF(pc). 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

D = {x € K : |x| < 1} .

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

(a) |x| = 0 if and only if x = 0;

(b) |xy| = |x||y| for all x,y € K;

(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 = pc 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

D* = D \ B = {x € K : |x| = 1} .

Clearly, D* is a group of units in K* and if x = 0, then can write x = pny,y € D*. Moreover, if U = {am : 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

x = ^ cl pl, with cl € U.

l=k

Recall that B is compact and open, so each fractional ideal

Bk = pkD = {x € K : |x| < q-k}

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, No and Z to denote the sets of natural, non-negative integers and integers, respectively.

Let x be a fixed character on K + that is trivial on D but non-trivial on B-1. Therefore, x is constant on cosets of D so if y € Bk, then xy(x) = x(y, x), x € K. Suppose that xu is any character on K+, then the restriction xu|D is a character on D. Moreover, as characters on D,xu = xv if and only if u — v € D. Hence, if {u(n) : n € N0} is a complete list of distinct coset representative of D in K+, then, as it was proved in [25], the set {xu(n) : n € N0} of distinct characters on D is a complete orthonormal system on D.

We now impose a natural order on the sequence {u(n)}^=0. We have D/B = GF(q) where GF(q) is a c-dimensional vector space over the field GF(p). We choose a set

{1 = Zo,Ci,C2,...,Cc-i}c D

~ n T?(

j }j=0

such that span {Zj }c=0 = GF(q). For n € No satisfying

0 < n < q, n = a0 + a1p + ••• + ac-1pc 1, 0 < ak < p, k = 0,1,...,c — 1,

we define

u(n) = (ao + aiZi +-----h ac-iZc-i) p .

Also, for

n = b0 + biq + b2q2 +-----h n € N0, 0 < bk < q, k = 0,1, 2,...,s,

we set

u(n) = u(b0) + u(bi)p-1 +-----h )p

This defines u(n) for all n € N0. In general, it is not true that u(m + n) = u(m) + u(n). But, if r, k € N0 and 0 < s < qk, then

u(rqk + s) = u(r)p-k + u(s). Further, it is also easy to verify that u(n) = 0 if and only if n = 0 and

{u(l) + u(k) : k € N0} = {u(k) : k € N0}

for a fixed I € N0. Hereafter we use the notation Xn = Xu(n), n > 0.

Let the local field K be of characteristic p > 0 and Zo, Z1, Z2, ..., Zc-1 be as above. We define a character x on K as follows:

x(C.P-j) = {

exp(2ni/p), ß = 0 and j = 1, 1, ß = 1,..., c — 1 or j = 1.

2.2. Fourier transforms on local fields

The Fourier transform of f € L:(K) is denoted by f({) and defined by

?{№)} = №)= ! f{x)xdx)dx. JK

'K

It is noted that

f (0 = f (x) X?(x)dx = f (x)x(—£x) dx.

JK JK

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 L:(K) into L^(K), and ||f < ||f ||1.

• If f € L1(K), then f is uniformly continuous.

• If f € L1(K) n L2(K), then ||f ||2 = ||f ||2.

The Fourier transform of a function f € L2(K) is defined by

f (£) = lim fk(0 = lim / f (x)x?(x) dx

k—œ k—^^ ,/|x|<qfc

where fk = f and $k is the characteristic function of Bk. Furthermore, if f € L2(D), then we define the Fourier coefficients of f as

f{u(n)) = / f{x)Xu(n){x)dx.

Jd

The series

X) f (u(n^ Xu(n) (x)

is called the Fourier series of f. From the standard L2-theory for compact Abelian groups, we conclude that the Fourier series of f converges to f in L2(D) and Parseval's identity holds:

/1|2 = |f(x)|2dx = |f (u(n))

raeNo

3. 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

and

An = {u(m)N + pu(j) : m € Z, 0 < j < N — 1} ,

where

Z = {u(n) : n € N0} .

It is easy to verify that A is not a group on local field K, but is the union of Z and a translate of Z.

2

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 {Vj : j € Z} of L2(K) such that the following properties hold:

(a) Vj C Vj+1 for all j € Z;

(b) [J,.eZ Vj is dense in L2(K);

(c) rijez Vj = {0};

(d) f (■) € Vj if and only if f (p-1 N■) € Vj+1 for all j € Z;

(e) There exists a function ^ in V0 such that {^>(- — A) : A € A}, is a complete orthonormal basis for Vo.

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-1N and |p-1| = q ensures that qNA C Z C A. For every j € Z, define Wj to be the orthogonal complement of Vj in Vj+1.

Then we have

Vj+1 = Vj ® Wj and We ± We if I = I'.

It follows that for j > J,

j-j-1

Vj = Vj ® 0 Wj-i,

i=o

where all these subspaces are orthogonal. By virtue of condition (b) in the Definition 2, this implies

L2(K)= 0 Wj,

j eZ

a decomposition of L2(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 A and dilations by the integral powers of p-1N form an orthonormal basis for L2(K).

Let a and b be any two fixed elements in K. Then, for any prime p and m, n € N0, let Dp, Tu(n)a and Eu(m)b be the unitary operators acting on f € L2(K) defined by:

Tu(n)af (x) = /(x — u(n)a), (Translation), Eu(m)b f (x) = x(u(m)bx)f (x), (Modulation), DPf(x) = y/qNf (p-1^) , (Dilation).

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Then for any f € L2(K), the following results can easily be verified:

F{Tu(n)af (x)} = E-u(n)a J{f (x) } , F{ Eu

(m )b f (x)} Tu(m )bF{ f (x)},

F{Dpj f (x)} = Dp-j F{f (x)},

DpjTu(n)a = T(qN)-ju(n)aDpj .

We state the following lemmas which will be very useful in establishing the results and whose proof can be found in [23].

Lemma 1. For an integer N > 1 and an odd integer r with 1 < r < qN — 1 .such that r and N are relatively prime. Let ^ € L2(K) with ||^>||2 = 1, then

(i) the family — A) : A € A} is an orthonormal system for fixed r if and only if

^\<p (£ + pu(k))|2 = q a.e. £ € K

fceNo

and _

£x(^W)l£(£ + P«(*0)|2 = O a.e. CeK;

fceN0 ^ '

(ii) the family — A) : A € AN} is an orthonormal system for every odd integer r if and only if

— Y)\2 = 1, a.e. £ € K. Lemma 2. Let (Vj, be non-uniform multiresolution analysis, where

Vo = span {ip(x — A) : A € A}. Then the necessary and sufficient condition for the existence of associated wavelets is

E — Y)\2 = 1 a.e. £ € K.

ysAn

Lemma 3. Let Sc K be measurable and Ao = {0,u(a)} + Z. Then (S, Ao) is a spectral pair if and only if there exist an integer N > 1 and an odd integer r with 1 < r < qN — 1, such that N and r are relatively prime , a = r/N and

N-1

5i/2 * 5nN * = 1.

j=0 nSN0

4. 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.

Theorem 1. A nonzero function ^ € L2(K) is a scaling function for wavelet NUMRA if and only if the following conditions are satisfied

(i) £ — Y)\2 = 1 a.e. £ € K;

tsAn

(ii) lim |^(p-1 N)jC|2 = 1 a.e. £ € q2D;

j ^^

(iii) there exist functions m1(£),m2(£) locally integrable, q-periodic functions such that

^(p-1 N£) = m(£M£) a.e. £ € K,

where

Proof. Suppose € L2(K) is a scaling function for wavelet NUMRA, say {Vj, ^}jeZ. Then by Lemma 2, we must have

E — Y)|2 = 1 a.e. { € K. (4.1)

ysAn

This gives (i). Since ^ € V0, we have Dp-i^ € V-1 C V0. Thus we can write

Dp-i ^ = E aATA^.

AeA

Taking the Fourier transform of both sides, we get

Dp^ = E aA^-A^.

AeA

So we can write,

^(p-1NY) = w-CYMY^

where

m(7) = m1(7)

and m1,™2 are q- periodic and locally integrable functions. This proves (iii).

Next we show that (ii) holds. Let f € L2(K) be such that /(7) = $^2^(7). Then

|2 _ 11 ill2

q-

As (Vj is NUMRA so if Pj is orthogonal projection onto Vj-, we must have

11/ - Pj/1|2 ^ 0 as j ^œ.

That is

l|Pj / ll^ll/1| as j ^œ.

Since |TA^>}AeA is an orthonormal bases for V0 so {DpjTA^>}AeA is an orthonormal basis for Vj. Thus

= a-e" 3^oo (4.2)

AeA q

E K/>DpjTa^)|2 = E K/.DpjTA^)|2 + E K/>DpjTA^)|2

AeA Aez Ae(u(r)/N+Z)

= E l/>DÄI2 + E l/>DÄI2

AeZ Ae(u(r)/N+Z)

- 2

fceNo

fceNo

.(p-1NV(p-1 N)j, 7 /t/,(r) (i\W~( 7~

2

E

keNo

= So I UqN)~J/2xu{k) ((rw) *(rw)d7

UqN)~j/2x«k) (rW V N

Y

(p-1N )j

dY

Putting

we obtain

Y

(qN )j

= n

AeA ^ keNo (p

+ E

keNo

(qN )j

keNo "(p-1n)-jD -i

2

' (p-1N D

\/9X«(fe)(P vMv)dv

2

E

keNo

' q2D

$ (p -1 jv) - i ® fe) (P v)viv)dv

+ E

keNo

Zq2 D

&{p-iN)-i<BX(-jï-v)VqXu(k)(P lr})(p{r})dr]

because (p 1N) jD C q2D, for any j > 0. Therefore from (4.2) and from the fact that {\/9Xit(fc)(P_1?7)} is an orthonormal basis for L2(qTi), we get

£ \(f,DviTx<p)\2 = (qNY [ MiDfdri - j oo.

Putting ß = (qN)j n, we get

Let

f 1iV)Jß|2dß ->• - as j ->• oo.

q2D q

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h(£) = lim |^(p-1 N)j£|2.

(4.3)

Then

0 < h(£) < 1 a.e. £ € q2D. Indeed for any fixed j € Z by using (4.1), we have

0 < |^(p-1N)j£|2 < 1 a.e. £ € q2D.

This gives

Now invoking the Lesbesgue-dominated convergence theorem, we obtain

0 < h(£) = lim |^(p-1N)j£|2 < 1 a.e. £ € q2D. j ^^

lim [ \fi((p-lN)j)ß\2dß= [ lim |£((p~lN)j)ß\2dß =

j'^^.Jq2D Jq2D q

Thus

[ HO<% = -= f 1 d£.

Jq2D q ./q2D

2

2

2

2

That is

f (1 — h(£))d£ = 0,

so by using

0 < h(£) < 1 a.e. £ € q2D,

we get h(£) = 1 a.e. £ € q2D. Hence (ii) is proved.

Conversely, let ^ € L2(K) satisfying (i)-(iii). We define closed subspaces Vj of L2(K) in the following way. { }

For j = 0 let Vj = span {^?(£ - A) : A € A} and for j / 0 let Vj = {/ : f((p~lN)-^) e Vo} • We will show (Vj, forms wavelet NUMRA. Using Lemma 1, the sequence {TA^}AeA is an orthonormal basis for V0.

By definition of Vj-, it can be easily shown that f (y) € Vj- if and only if

f ((p-1 N)y) € Vj-+1,

which clearly implies P|jeZ Vj- = {0}. To prove Vj- C Vj+1, it is sufficient to show that V0 C V1. First we show that

Vj = {fe L2(K) : /((p-^Y) = (m)(7) + X ( ^7 W(7))£(7)¡7 (4-4)

where mj1,m2 are locally integrable, q-periodic functions. Let f € Vj-, then

1

rDp-j f (Y) € Vo,

(qN)j/2"p

as |TA^}AeA is an orthonormal basis for V0, so there exist {j € l2(N0) such that On taking Fourier transform of both sides, we obtain

/((p-^Y) = ecjaxa(P"1Y)^(Y) = {'"!•:-) • \ (^Y)m2(7)W(Y),

AeA ^ \ J )

where ml and m2 are locally integrable and q-periodic functions. If f € L2(K) satisfies

/((p-^Y) = { m)(7) +X ('^7 K2(Y) ^(7) for some ml and m2 are locally integrable and q-periodic functions, then we can write

'u(r)

jr',

fcez x 7 fceNo

for some scalars {ck} and {dk}fceN0 € I2(No). Therefore

Dp> m

(qN)j/2 = EFAX«(fc)(P_17)^(7)

for some {jJAeA € l2(N0). By taking inverse Fourier transform on both sides, we obtain

where

/((p-1N )j y) = £

AeA

£ K|2 < ^

AeA

which shows f (7) € Vj. Hence Vj(j € Z) are given by (4.4).

Now we are ready to show that V0 C V1. Let f (7) € V0. Then by (4.4), we can write

/(7) = {m¿(7) + X (^y)™¿(YMY)},

where m¿ and m0 are locally integrable, q-periodic functions. Therefore,

/(y) = G(Y )m(Y )^(Y),

(4.5)

where and

This gives

G{l) = mlip-'N!) +x«(r)(p-17K(p-1iV7)

-1 2 -1

m(7) = m1(7) +x ( ^r^Y )™2(y)-

N

G(7)m(7) = G(7){m1(7) +X (^y)™2(Y)} = G(7)m1(7) +X (^Y)G(7)m2(7). (4-6)

Using the conditions (i) and (iii), it can be easily shown that functions m1(7) and m2(7) are bounded. Also since m1(7), m2(7) and G(7) are q-periodic, therefore the functions G(7)m1(7) and G(y)m2 (7) are q-periodic and

/ |G(Y)m1 (7)|2d7, / |G(Y)m2(7)|2d7 < œ.

./D ./D

Thus by using (4.4)-(4.6), we infer that /(7) € V\. Hence V0QVi.

To prove that IJ;. Vo = L2(K), it sufficient to show that, for any / € L2(K), we have

IIP f — f Il2 = 11/Il2 — l|P4.12j f ||2 ^ 0 as j ^œ,

where Pj is the orthonormal projection onto Vj. Let f € L2(K) be such that f € Cc(K). Now we have

IIPf II2 = E Kf,D„jTA^)|2 = E Kf,Dp^)|2

AeA

E f (qN)-1/2/(y)x„(fc)

./K

keNo

Y

AeA

u(r)

(p-1N)J V N Y Z' u(r)

+ pu(k) (

Y

(p-1N )J

dY

+ EI /:(^r1/2/>,)x„« (t1+m«)) ?

K

(4.7)

keNo

keNo

+ E / (^^/((P'WOxWP-1^)^

K

2

2

Since / has compact support, we can choose j so large that

supp/((p-1 N)j£) Ç q2D. Then, using the fact that {s/qXu{k)(0} is an orthonormal basis for li2(q2Q) and by (4.7), we get

I w> - E

^ fceNo

/ fdp-'NyovQXuwip-'omdc

Zq2 D

+ £

fceNo

/ mp-'NyOVqe^Xuwip-'vMv)^

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' q2D

(4.8)

= (gNy / im-1 Nmmfdt

J q2D

Putting (p-1N)j£ = n in (4.8) and invoking the Lesbesgue-dominated convergence theorem, we

get

IIPf II2 =/ .f(n)^(P-1N)-jn dn ^IIfII2 as j ^rc.

./(p-1N )-j D

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.

Theorem 2. If ( be a .scaling function of wavelet NUMRA and ( is continuous then |(?(0)| = 1 and ( (u(m)N — u(j)) = 0, where m € No, 0 < j < N — 1. In particular ((u(m)N) = 0 for m € N0 and ((—pu(j)) =0, 0 < j < N — 1.

Proof. By (4.3), we have

lim [ \fi{p-lN)-jß\2dß = -

q

as is continuous. By virtue of Lebesgue dominated convergence theorem, we obtain |((0)| = 1. Since ( is a scaling function for wavelet NUMRA, we have

E !£(£ — Y)|2 = 1 a.e. £ € K. (4.9)

ysAn

Suppose

( (u(m)N — pu(j)) = a = 0 for some m, j not both zero together. Then

l^(£)l + l( (£ + u(m)N — pu(j))|2 > 1 + a2, when £ € peD

for some e > 0 which contradicts (4.9). □

The following theorem gives the sufficient conditions for the frequency band of the scaling function of wavelet NUMRA on LFPC.

2

2

2

Theorem 3. Let 0 be a compact subset of K such that

(i) U C (p-1N)U;

(ii) UmgNo (P-1N)jU = K;

N -1

(iii) £ ¿j/2 * £ ^mN * = 1.

j=0 mSN0

Then U is the frequency band function for some wavelet NUMRA.

Proof. Let

Vj = {f € L2(K) : supp f C (p-1 N)jU, j € z}

and ^ € L2(K) be such that (p = Using hypothesis (i) and the definition of Vj, we have Vj C Vj+1 and f ((p-1N)j7) € Vj if and only if f((p-1N)j+1Y) € Vj+1. By hypothesis (ii) and the definition of Vj, we get UjgZ V = L2(K). By using Lemma 3 and hypothesis (iii), we get that (U, A) is a spectral pair. Now we have

and the Fourier transform is the unitary operator. Thus {TA^>}AeA is an orthonormal basis for V0. By virtue of Lemma 3, we infer that P|jeZ Vj = {0}. Hence 0 is frequency band for wavelet NUMRA (Vj □

5. 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 L2(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.

Acknowledgements

The authors pay gratitude to the referees for their valuable suggestions and comments.

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