Complex Anal. Oper. Theory https://doi.org/10.1007/s11785-018-0813-6

Complex Analysis and Operator Theory

Nonuniform Discrete Wavelets on Local Fields of Positive Characteristic M. Younus Bhat1

Received: 23 October 2017 / Accepted: 5 June 2018 © Springer International Publishing AG, part of Springer Nature 2018

Abstract A constructive algorithm based on the theory of spectral pairs for constructing nonuniform wavelet basis in L 2 (R) was considered by Gabardo and Nashed (J Funct Anal 158:209–241, 1998). In this setting, the associated translation set is a spectrum  which is not necessarily a group nor a uniform discrete set, given  = {0, r/N } + 2 Z, where N ≥ 1 (an integer) and r is an odd integer with 1 ≤ r ≤ 2N − 1 such that r and N are relatively prime and Z is the set of all integers. The objective of this paper is to develop nonuniform discrete wavelets on local fields. We first provide a characterization of an orthonormal basis for the Hilbert space l 2 (λ) and then show that it can be expressed as orthogonal decomposition in terms of countable number of its closed subspaces. Moreover, we show that the wavelets associated with NUMRA on local fields of positive characteristic are connected with the wavelets on spectrum. Keywords Wavelet · Nonuniform · Discrete wavelet · Local field · Fourier transform Mathematics Subject Classification 42C40 · 42C15 · 43A70 · 11S85

Communicated by Sanne ter Horst, Dmitry Kaliuzhnyi-Verbovetskyi, Izchak Lewkowicz and Daniel Alpay.

B 1

M. Younus Bhat [email protected] Department of Mathematics, Islamic University of Science and Technology, Awantipora, Jammu and Kashmir 192122, India

M. Y. Bhat

1 Introduction Multiresolution analysis (MRA) is an important mathematical tool since it provides a natural framework for understanding and constructing discrete wavelet  systems. A multiresolution analysis is an increasing family of closed subspaces V j : j ∈ Z   of L 2 (R) such that j∈Z V j = {0} , j∈Z V j is dense in L 2 (R) and which satisfies f ∈ V j if and only if f (2·) ∈ V j+1 . Furthermore, there exists an element ϕ ∈ V0 such that the collection of integer translates of function ϕ, {ϕ(· − k) : k ∈ Z} represents a complete orthonormal system for V0 . The function ϕ is called the scaling function or the father wavelet. The concept of multiresolution analysis hasbeen  extended in various ways in recent years. These concepts are generalized to L 2 Rd , to lattices different from Zd , allowing the subspaces of multiresolution analysis 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 ∈ G L d (R) as long as A ⊂ AZd . Recently, Gabardo and Nashed [6–8] developed the theory of nonuniform wavelets and wavelet sets in L 2 (R) for which the translation set is no longer a discrete subgroup of R, but a union of two lattices, which is associated with a famous open conjecture of Fuglede on spectral pairs. For more about wavelets and their applications, we refer the monographs [1,4,20]. In recent years there has been a considerable interest in the problem of constructing wavelet bases on various groups, namely, Cantor dyadic groups [11], locally compact Abelian groups [5], p-adic fields [10] and Vilenkin groups [12]. Recently, Benedetto and Benedetto [2] developed a wavelet theory for local fields and related groups. They did not develop the multiresolution analysis (MRA) approach, their method is based on the theory of wavelet sets and only allows the construction of wavelet functions whose Fourier transforms are characteristic functions of some sets. Since local fields are essentially of two types: zero and positive characteristic (excluding the connected local fields R and C). Examples of local fields of characteristic zero include the p-adic field Q p where as local fields of positive characteristic are the Cantor dyadic group and the Vilenkin p-groups. Even though the structures and metrics of local fields of zero and positive characteristics are similar, but their wavelet and multiresolution analysis theory are quite different. The concept of multiresolution analysis on a local field K of positive characteristic was introduced by Jiang et al. [9]. They pointed out a method for constructing orthogonal wavelets on local field K with a constant generating sequence. Subsequently, tight wavelet frames on local fields of positive characteristic were constructed by Shah and Debnath [19] using extension principles. More results in this direction can also be found in [3,13–19] and the references therein. Recently, Shah and Abdullah [15] 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 ∈ N0 } 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  = {0, u(r )/N }+Z, where

Nonuniform Discrete Wavelets on Local Fields of Positive…

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. Motivated and inspired by the concept of nonuniform multiresolution analysis on local fields of positive characteristic and wavelets on spectrum [21], we construct the associated discrete wavelets on local fields of positive characteristic. More precisely, We first provide a characterization of an orthonormal basis for l 2 () and then show that the Hilbert space l 2 () can be expressed as an orthogonal decomposition in terms of countable number of its closed subspaces. Moreover, we show that the wavelets associated with NUMRA are connected with the wavelets on local field. This paper is organized as follows. In Sect. 2, we discuss some preliminary facts about local fields of positive characteristic and also some results which are required in the subsequent sections including the definitions of uniform and non-uniform multiresolution analysis on local fields of positive characteristic. In Sect. 3, we provide a characterization of an orthonormal basis for l 2 () in the form of unitary matrix. In Sect. 4, we show that the Hilbert space l 2 () can be expressed as an orthogonal decomposition of countable number of its closed subspaces. In Sect. 5, we provide a connection between the wavelets associated with NUMRA and the wavelets on local fields of positive characteristic.

2 Preliminaries on Local Fields Let K 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. Further, 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, non-discrete and totally disconnected. The p-adic fields are examples of local fields. More details are referred to [22]. In the rest of this paper, we use the symbols N, N0 and Z to denote the sets of natural, non-negative integers and integers, respectively. Let K be a local field. Let d x be the Haar measure on the locally compact Abelian group K + . If α ∈ K and α = 0, then d(αx) is also a Haar measure. Let d(αx) = |α|d x. 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) |x y| = |x||y| for all x, y ∈ K ; and (c) |x + y| ≤ max {|x|, |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 hence as result B is both principal and prime. Since the local field K is totally disconnected, so there exist an element of B of maximal absolute value. Let p be a fixed element of maximum absolute value in B. Such an element is called a prime element of K . Therefore, for such an ideal B in D, we have B = p = pD. As it was proved in [22], 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 G F(q), where q = p k for some prime p and k ∈ N.

M. Y. Bhat

Let D∗ = 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 = pk x  , x  ∈ D∗ . For a proof of this fact we refer to [22]. Moreover, each Bk = pk D = x ∈ K : |x| < q −k is a compact q−1 subgroup of K + and usually known as the fractional ideals of K + . Let U = {ai }i=0 be any fixed full set of coset representatives of B in D, then every element x ∈ K can   be expressed uniquely as x = ∞ =k c p with c ∈ U. Let χ be a fixed character on K + that is trivial on D but is non-trivial on B−1 . Therefore, χ is constant on cosets of D so if y ∈ Bk , then χ y (x) = χ (yx), x ∈ K . Suppose that χu is any character on K + , then clearly the restriction χu |D is also a character on D. Therefore, if {u(n) : n ∈ N0 } is a completelist of distinct coset representative of D in K + , then, as it was proved in  [22], the set χu(n) : n ∈ N0 of distinct characters on D is a complete orthonormal system on D. The Fourier transform fˆ of a function f ∈ L 1 (K ) ∩ L 2 (K ) is defined by fˆ(ξ ) =

K

f (x)χξ (x)d x.

(2.1)

It is noted that fˆ(ξ ) =



K

f (x) χξ (x)d x =

f (x)χ (−ξ x)d x. K

Furthermore, the properties of Fourier transform on local field K are much similar to those of on the real line. In particular Fourier transform is unitary on L 2 (K ). ∼ We now impose a natural order on the sequence {u(n)}∞ n=0 . We have D/B = G F(q) where G F(q) is a c-dimensional vector space over the field G F( p). We choose a set  c−1 {1 = ζ0 , ζ1 , ζ2 , . . . , ζc−1 } ⊂ D∗ such that span ζ j j=0 ∼ = G F(q). For n ∈ N0 satisfying 0 ≤ n < q, n = a0 + a1 p + · · · + ac−1 p c−1 , 0 ≤ ak < p, and k = 0, 1, . . . , c − 1, we define u(n) = (a0 + a1 ζ1 + · · · + ac−1 ζc−1 ) p−1 .

(2.2)

Also, for n = b0 + b1 q + b2 q 2 + · · · + bs q s , n ∈ N0 , 0 ≤ bk < q, k = 0, 1, 2, . . . , s, we set u(n) = u(b0 ) + u(b1 )p−1 + · · · + u(bs )p−s .

(2.3)

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 < q k , then u(rq k + 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() + u(k) : k ∈ N0 } = {u(k) : k ∈ N0 } for a fixed  ∈ N0 . Hereafter we use the notation χn = χu(n) , n ≥ 0. Let the local field K be of characteristic j > 0 and ζ0 , ζ1 , ζ2 , . . . , ζc−1 be as above. We define a character χ on K as follows:

Nonuniform Discrete Wavelets on Local Fields of Positive…

χ (ζμ p− j ) =



exp(2πi/j), 1,

μ = 0 and j = 1, μ = 1, . . . , c − 1 or j = 1.

(2.4)

Let us recall the definition of an MRA on local fields of positive characteristic [9]. Definition 2.1 Let K be a local field of positive characteristic p > 0 and p be a prime element of K . A multiresolution analysis(MRA) of L 2 (K ) is a sequence of closed subspaces {V j : j ∈ Z} of L 2 (K ) satisfying the following properties: (a) (b) (c) (d) (e)

V j ⊂ V j+1 for all j ∈ Z;  2  j∈Z V j is dense in L (K ); j∈Z V j = {0}; f (·) ∈ V j if and only if f (p−1 ·) ∈ V j+1 for all j ∈ Z;    Thereis a function ϕ ∈ V0 , called the scaling function, such that ϕ · −u(k) : k ∈ N0 forms an orthonormal basis for V0 .

According to the standard scheme for construction of MRA-based wavelets, for each j, we define a space W j (wavelet space) as the orthogonal complement of V j in V j+1 , i.e., V j+1 = V j ⊕ W j , j ∈ Z, where W j ⊥ V j , j ∈ Z. It is not difficult to see that f (·) ∈ W j if and only if f (p−1 ·) ∈ W j+1 ,

j ∈ Z.

(2.5)

Moreover, they are mutually orthogonal, and we have the following orthogonal decompositions:

L 2 (K ) =

 j∈Z

⎛ W j = V0 ⊕ ⎝



⎞ Wj⎠ .

(2.6)

j≥0

As in the case of Rn , we expect the existence of q − 1 number of functions ψ1 , ψ2 , . . . , ψq−1 to form a set of basic wavelets. In view of (2.5) and (2.6),  if ψ1 , ψ2 , . . . , ψq−1 is a set of functions such that the system it is clear that ψ · −u(k) : 1 ≤  ≤ q − 1, k ∈ N0 forms an orthonormal basis for W0 , then  j/2   q ψ (p− j x − u(k) : 1 ≤  ≤ q − 1, j ∈ Z, k ∈ N0 forms an orthonormal basis for L 2 (K ). Let Z = {u(n) : n ∈ N0 }, where {u(n) : n ∈ N0 } is a complete list of (distinct) coset representation of D in K + . For an integer N ≥ 1 and an odd integer r with 1 ≤ r ≤ q N − 1 such that r and N are relatively prime, we define 

u(r ) + Z.  = 0, N It is easy to verify that  is not a group on local field K , but is the union of Z and a translate of Z. We are now in a position to define a nonuniform multiresolution analysis (NUMRA) on local fields of positive characteristic as follows:

M. Y. Bhat

Definition 2.2 For an integer N ≥ 1 and an odd integer r with 1 ≤ r ≤ q N − 1 such that r and N are relatively prime, an associated nonuniform multiresolution  analysis 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: (a) (b) (c) (d) (e)

V j ⊂ V j+1 for all j ∈ Z;  2  j∈Z V j is dense in L (K ); j∈Z V j = {0}; f (·) ∈ V j if and only if f (p−1 N ·) ∈ V j+1 for all j ∈ Z; There exists a function ϕ in V0 such that {ϕ(· − λ) : λ ∈ }, is a complete orthonormal basis for V0 .

It is worth noticing that, when N = 1, one recovers from the definition above the definition of a multiresolution analysis 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 q N  ⊂ Z ⊂ . For every j ∈ Z, define W j to be the orthogonal complement of V j in V j+1 . Then we have V j+1 = V j ⊕ W j and W ⊥ W if  =  .

(2.7)

It follows that for j > J , V j = VJ ⊕

j−J −1

W j− ,

(2.8)

=0

where all these subspaces are orthogonal. By virtue of condition (b) in the Definition 2.2, this implies L 2 (K ) =



Wj,

(2.9)

j∈Z

a decomposition of L 2 (K ) into mutually orthogonal subspaces. As in the standard case, one expects the existence of q N − 1 number of functions so that their translation by elements of  and dilations by the integral powers of p−1 N form an orthonormal basis for L 2 (K ).   Definition 2.3 A set of functions ψ1 , ψ1 , . . . , ψq N −1 in L 2 (K ) is saidto be a set of basic wavelets associated with the nonuniform multiresolution analysis V j : j ∈ Z if the family of functions {ψ (· − λ) : 1 ≤  ≤ q N − 1, λ ∈ } forms an orthonormal basis for W0 . Let us define the spaces 

  2 z(λ) < ∞ l () = z :  → C : 2

λ∈

Nonuniform Discrete Wavelets on Local Fields of Positive…

and

L ( ) =

f : →C:

2



    f (ξ )2 dξ < ∞ ,

where is a Lebesgue measurable subset of K with finite positive measure. These spaces are Hilbert spaces with the inner product defined by 

z, w =

z(λ)w(λ) for z, w ∈ l 2 ()

λ∈

and f, g =



f (ξ )g(ξ )dξ for f, g ∈ L 2 ( ),

respectively. Definition 2.4 The Fourier transform on l 2 () is a map ∧ : l 2 () → L 2 ( ) defined by zˆ (ξ ) =



z(λ)χλ (ξ ), z ∈ l 2 (λ)

λ∈

and its inverse is   f ∨ (λ) = f, χλ (ξ ) =



f (ξ )χλ (ξ ) dξ,

f ∈ L 2 ( ).

For all z, w ∈ l 2 (), we have the Parseval’s identity; z, w =



z(λ)w(λ) =



λ∈

f (ξ )g(ξ )dξ = ˆz , w , ˆ

and the Plancherel’s identity;    2      z  = z(λ)2 = zˆ (ξ )2 dξ = zˆ 2 .

λ∈

For λ ∈ , we define the translation operator Tq N λ : l 2 () → l 2 () by Tq N λ z(σ ) = z(σ − q N λ), ∀ σ ∈ . Then, it can be easily verified that for z, w ∈ l 2 (), 

Tq N λ z

∧

(ξ ) = χq N λ (ξ )ˆz (ξ ) and



   Tq N λ z, Tq N σ w = Tq N (λ−σ ) z, w .

M. Y. Bhat

3 Characterization of an Orthonormal Basis for l 2 () In this section, we provide a characterization of an orthonormal basis for l 2 () using the translation operator.     Theorem 3.1 For z, w ∈ l 2 (), the systems Tq N λ z λ∈ and Tq N λ w λ∈ generates orthogonal subspaces in l 2 () if and only if the following conditions hold: (a)

q N −1 s=0

     u(s) u(s) wˆ ξ + −1 zˆ ξ + −1 p N p N     u(s) u(s) + zˆ ξ + −1 + u(N ) wˆ ξ + −1 + u(N ) = 0, p N p N

(b)

q N −1 s=0

     u(s) u(s) pu(s) zˆ ξ + −1 χ wˆ ξ + −1 N p N p N r



    u(s) u(s) + zˆ ξ + −1 + u(N ) wˆ ξ + −1 + u(N ) = 0, p N p N and (c)

q N −1 s=0

     u(s) u(s) pu(s) zˆ ξ + −1 χ wˆ ξ + −1 N p N p N r



    u(s) u(s) + zˆ ξ + −1 + u(N ) wˆ ξ + −1 + u(N ) = 0. p N p N   Proof The orthogonality of the subspaces generated by the systems Tq N λ z λ∈ and   Tq N λ w λ∈ , for z, w ∈ l 2 (), with λ = σ , is equivalent to   0 = Tq N λ z, Tq N σ w  ∧  ∧  = Tq N λ z , Tq N σ w = zˆ (ξ )w(ξ ˆ ) χp−1 N (λ−σ ) (ξ )dξ   = zˆ (ξ )w(ξ ˆ ) + zˆ (ξ + u(N )) wˆ (ξ + u(N )) χp−1 N (λ−σ ) (ξ )dξ λ, σ ∈ . pD

(3.1) Taking λ = u(m) and σ = u(n) where m, n ∈ N0 , in Eq. (3.1) and setting h(ξ ) = zˆ (ξ )w(ξ ˆ ) + zˆ (ξ + u(N )) wˆ (ξ + u(N )),

(3.2)

Nonuniform Discrete Wavelets on Local Fields of Positive…

we have

    h(ξ ) χ (p−1 N )u q(m − n) ξ dξ pD ⎧ ⎫  N −1  ⎨q    u(s) ⎬  −1 h ξ + −1 χ (p N )u q(m − n) ξ dξ. = p N ⎭ (p/q N )D ⎩

0=

s=0

As the above equality holds for all (m − n) ∈ N0 , it follows that q N −1 s=0

  u(s) = 0, a.e. h ξ + −1 p N

(3.3)

On taking λ = r/N + u(m) and σ = r/N + u(n) where m, n ∈ N0 , we will obtain the same identity (3.3). Similarly, taking λ = r/N + u(m) and σ = u(n), where m, n ∈ N0 , in (3.1), we obtain

    h(ξ ) χ (p−1 N )u q(m − n) ξ χ (p−1r ξ ) dξ pD ⎧ ⎫  N −1  ⎨q   u(s) ⎬ r pu(s) h ξ + −1 χ = N p N ⎭ (p/q N )D ⎩

0=

s=0

    × χ (p−1 N )u q(m − n) ξ χ (p−1r ξ ) dξ. Hence, we conclude that q N −1 s=0

    u(s) r h ξ + −1 χ pu(s) = 0, a.e. p N N

(3.4)

Taking λ = u(m) and σ = r/N + u(n), we have λ − σ = −r/N + u(m − n), where m, n ∈ N0 , we deduce that q N −1 s=0

    r u(s) χ pu(s) = 0, a.e. h ξ + −1 p N N

This completes the proof of the Theorem.

(3.5)  

As the collection {χλ (ξ )}λ∈ is an orthonormal basis for L 2 ( ), there exist locally functions z 1 , z 2 , w1 and w2 such that

L 2 , p-periodic

zˆ (ξ ) = z 1 (ξ ) + χ

r  r  ˆ ) = w1 (ξ ) + χ ξ z 2 (ξ ), and w(ξ ξ w2 (ξ ). (3.6) N N

M. Y. Bhat

Hence equation (3.2) becomes   h(ξ ) = q z 1 (ξ )w1 (ξ ) + z 2 (ξ )w2 (ξ ) .

(3.7)

Therefore, the Eqs. (3.3)–(3.6) yield 

    u(s) u(s) w1 ξ + −1 z 1 ξ + −1 p N p N s=0     u(s) u(s) + z 2 ξ + −1 w2 ξ + −1 = 0, p N p N      q N −1   u(s) u(s) r pu(s) z 1 ξ + −1 χ w1 ξ + −1 N p N p N s=0      u(s) u(s) + z 2 ξ + −1 w2 ξ + −1 = 0, p N p N q N −1

(3.8)

(3.9)

and 

    u(s) u(s) χ w1 ξ + −1 pu(s) z 1 ξ + −1 N p N p N s=0      u(s) u(s) + z 2 ξ + −1 w2 ξ + −1 = 0. p N p N

q N −1

r



(3.10)

Corollary 3.2 Suppose for z, w ∈ l 2 (),(3.6) holds. Then the subspaces generated by the systems Tq N λ z λ∈ and Tq N λ w λ∈ are orthogonal in l 2 () if and only if the equations (3.8)–(3.10) hold.   Theorem 3.3 For z ∈ l 2 (), the system Tq N λ z λ∈ is orthonormal in l 2 () if and only if the following conditions hold:     2  q N −1    u(s) 2  u(s) 1   zˆ ξ + −1 + zˆ ξ + −1 + u(N )  = q N , (3.11)   q p N p N s=0

and q N −1 s=0

 2   2        u(s) u(s)  + zˆ ξ + pu(s) zˆ ξ + −1 + u(N )  = 0. χ   −1 N p N p N r

(3.12)

Nonuniform Discrete Wavelets on Local Fields of Positive…

  Proof For z ∈ l 2 (), and λ, σ ∈ , the orthonormality of the system Tq N λ z λ∈ in l 2 () is equivalent to           zˆ (ξ )2 + zˆ (ξ + u(N ))2 χ (p−1 N )u q(m − n) ξ dξ = δλ,σ . pD

Therefore, proceeding in a similar way as in the proof of Theorem 3.1, we obtain the required proof.     Corollary 3.4 Suppose for z ∈ l 2 (), (3.6) holds. Then the system Tq N λ z λ∈ is orthogonal in l 2 () if and only if the following equations hold. (a)

q N −1 s=0

(b)

q N −1 s=0

  2   2      u(s) u(s) zˆ ξ +  + zˆ ξ +  = q N,   p−1 N  p−1 N  χ

         u(s) 2  u(s) 2  pu(s) zˆ ξ + −1 + zˆ ξ + −1 = 0. N p N  p N 

r

Theorem 3.5 Let us define the set F by   F = Tq N λ wk : wk ∈ l 2 (); λ ∈ ; 0 ≤ k ≤ q N − 1 .

(3.13)

Then the following statements are equivalent: (i) The set F is an orthonormal basis for l 2 (). (ii) The matrix M(ξ ) of order q 2 N × q 2 N is unitary, when the entries Mst (ξ ), 0 ≤ s, t ≤ q 2 N − 1 of M(ξ ) are defined as follows: ⎧  ⎪ ⎪ wˆ t ξ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ w ˆ ξ t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎨  r Mst (ξ ) = √ χ q N ⎪ ⎪ N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ r ⎪ ⎪ ⎪ ⎪ χ ⎪ ⎪ N ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u(s) + −1 p N +

 ; 0 ≤ s ≤ q N − 1; 0 ≤ t ≤ q N − 1,

 u(s − q N ) + u(N ) ; q N ≤ s ≤ q 2 N − 1; p−1 N 0 ≤ t ≤ q N − 1,

   u(s − q N ) ; 0 ≤ t ≤ q N − 1; pu(s) wˆ t−q N ξ + p−1 N q N ≤ s ≤ q 2 N − 1,    u(s − q N ) pu(s) wˆ t−q N ξ + + u(N ) ; p−1 N q N ≤ s ≤ q 2 N − 1; q N ≤ t ≤ q 2 N − 1,

(3.14)

M. Y. Bhat

Proof Suppose that the system F defined by (3.13) is an orthonormal basis for l 2 (). Then, F is an orthonormal set in l 2 () and therefore, for λ, σ ∈ , and 0 ≤ , k ≤ q N − 1, we have 

 Tq N λ w , Tq N σ wk = δλ,σ δ,k .

Hence, for each  and k, 0 ≤ , k ≤ q N − 1, we have by Theorems 3.1 and 3.3      q N −1 u(t) u(t) 1  wˆ k ξ + −1 wˆ  ξ + −1 q p N p N t=0     u(t) u(t) + wˆ  ξ + −1 + u(N ) wˆ k ξ + −1 + u(N ) = q N δ,k , p N p N      q N −1   u(t) u(t) r χ wˆ k ξ + −1 R,k (ξ ) = pu(t) wˆ  ξ + −1 N p N p N t=0     u(t) u(t) + wˆ  ξ + −1 + u(N ) wˆ k ξ + −1 + u(N ) = 0, p N p N

(3.15)

(3.16)

and S,k (ξ ) =



    u(t) u(t) wˆ k ξ + −1 wˆ  ξ + −1 N p N p N t=0     u(t) u(t) + wˆ  ξ + −1 + u(N ) wˆ k ξ + −1 + u(N ) = 0. p N p N

q N −1

χ

r

pu(t)



(3.17)

Thus, it is sufficient to consider the Eqs. (3.15) and (3.16), as S,k (ξ ) = R,k (ξ ). These equations give rise to the matrix M(ξ ) of order q 2 N × q 2 N with entries as defined in system (3.14). Moreover, the matrix M(ξ ) is unitary. This follows from the Eqs. (3.15) 2 and (3.16) and noting that columns of M(ξ ) form an orthonormal system in Cq N 2 with respect to the usual inner product and hence form an orthonormal basis of Cq N . Conversely, assume that the matrix M(ξ ) is unitary. Then, it suffices to show that the set F is complete. For this, let w ∈ l 2 () and Pw be the projection onto span F, then we have q N −1  & &   & Pw &2 =  w, Tq N λ w 2 2 =0 λ∈

=

q N −1 

 2  w, ˆ (Tq N λ w )∧ 

=0 λ∈

=

q N −1   =0 λ∈

  



2  w(ξ ˆ )wˆ  (ξ )χλ (p−1 N ξ )dξ  .

Nonuniform Discrete Wavelets on Local Fields of Positive…

Writing = pD∪p(N +D) and observing that pD = & & & Pw &2 = 2

q N −1 s=0

p(s +D)/N , we get

 q N −1 q N −1      1  pu(t) wˆ  (ζt ) wˆ  (ζt ) χ λ  (q N )2 =0 λ∈  pD t=0 2    + wˆ (ζt + u(N )) wˆ  (ζt ) χλ (ξ )dξ  . 

(3.18)

  where ζt = ξ + u(t) /p−1 N . As  = Z ∪ {Z + u(r )/N }, we can rewrite (3.18), by using Plancherel formula, as follows  q N −1  q N −1    1  wˆ (ζs ) wˆ  (ζs )  (q N )2 =0 m∈N0  pD s=0 2       + wˆ (ζs ) wˆ  ζs + u(N ) χ pu(m)ξ dξ    q N −1  q N −1     1 r  pu(s) wˆ (ζs ) wˆ  (ζs ) + χ  (q N )2 N =0 m∈N0  pD s=0 2       r   ξ χ pu(m)ξ dξ  + wˆ (ζs + u(N )) wˆ  ζs + u(N ) χ N      q N −1 q N −1  u(s) 1   ξ + = 3 2 w ˆ w ˆ ) (ζ s   q N p−1 N =0  pD s=0 2      + wˆ ζs + u(N ) wˆ  ζs + u(N )  dξ   q N −1  q N −1   1   r + 3 2 pu(s) wˆ (ζs ) wˆ  (ζs ) χ  q N N =0  pD s=0 2       + wˆ ζs + u(N ) wˆ  ζs + u(N )  dξ. 

& & & Pw &2 = 2

As the rows of matrix M(ξ ) form an orthonormal system in Cq results for 0 ≤ r, s ≤ q N − 1:

2N

, we have following

M. Y. Bhat

    q N −1    r u(s) u(s) 1  1−χ pu(r ) − pu(s) wˆ  ξ + −1 wˆ  ξ + −1 q N p N p N =0

= q N δr,s   q N −1    r u(s) 1  1−χ pu(r ) − pu(s) wˆ  ξ + −1 + u(N ) q N p N =0   u(s) × wˆ  ξ + −1 + u(N ) = q N δr,s p N and q N −1



=0

1−χ



r

pu(r ) − pu(s)

N

wˆ 

    u(s) u(s) × ξ + −1 wˆ  ξ + −1 + u(N ) = 0. p N p N

Therefore for 0 ≤ s, t ≤ q N − 1, we have q N −1 q N −1  2 1   w ˆ w ˆ (ζ ) (ζ )   s  s (q N )2 pD =0 s=0  2    + wˆ (ζs + u(N )) wˆ  (ζs + u(N )) dξ ⎧ q N −1  N −1 ⎨ 2  q  1   wˆ (ζs )2 = w ˆ (ζ )    s ⎩ (q N )2 pD s=0 =0 ⎫ N −1    2 q 2 ⎬      + wˆ ζs + u(N )  wˆ  (ζs + u(N )) dξ ⎭

& & & Pw &2 = 2

=0

=

1 qN



q N −1 

pD s=0

      wˆ (ζs )2 + wˆ ζs + u(N ) 2 dξ

     2  q N −1   pξ 2  pξ 1   wˆ + u(N )  dξ = + wˆ   qN N N s=0 (1+s)pD      2  2    pξ pξ 1  + wˆ wˆ + u(N )  dξ =  q N ND  N  N        wˆ (ξ )2 + wˆ ξ + u(N ) 2 dξ = pD

Nonuniform Discrete Wavelets on Local Fields of Positive…

=



  wˆ (ξ )2 dξ

& &2 = &wˆ &2 & &2 = &w & . 2

Hence, projection P is an identity map and span F = l 2 (). Therefore, the set F   is an orthonormal basis for l 2 (). This completes the proof. Corollary 3.6 For each , 0 ≤  ≤ q N − 1, let w ∈ l 2 () such that wˆ  (ξ ) = w0 (ξ ) + χ

r  ξ w1 (ξ ), N

(3.19)

for some L 2 , p-periodic functions w0 and w1 . Then, the system (3.13) is an orthonormal basis for l 2 () if and only if the matrix ⎧   u(s) ⎪ ⎪ w ξ + ; 0 ≤ s ≤ q N − 1; 0 ≤ t ≤ q N − 1, ⎪ t0 ⎪ ⎪ p−1 N ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ u(s − q N ) ⎪ ⎪ ; q N ≤ s ≤ q 2 N − 1; 0 ≤ t ≤ q N − 1, ξ + w ⎪ t1 ⎪ −1 N ⎪ p ⎪ ⎪ ⎪ ⎪ ⎪   ⎪  ⎪ r u(s − q N ) 1 ⎨χ pu(s) w(t−q N )0 ξ + ; 0 ≤ t ≤ q N − 1; Mst (ξ ) = √ N p−1 N ⎪ qN ⎪ ⎪ ⎪ ⎪ ⎪ q N ≤ s ≤ q 2 N − 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪   ⎪  r ⎪ u(s − q N ) ⎪ ⎪ ξ + pu(s) w ; q N ≤ s ≤ q 2 N − 1; χ ⎪ (t−q N )1 ⎪ −1 N ⎪ N p ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ q N ≤ t ≤ q 2 N − 1, is unitary. Definition 3.7 The system F defined by (3.13) is called first-stage nonuniform discrete wavelet system for l 2 () if F is a complete orthonormal setin l 2 (). we called  w0 as the nonuniform father wavelet and wk : 1 ≤ k ≤ q N − 1 as the nonuniform mother wavelet. Assume w0 in l 2 () such that it satisfies Eqs. (3.11) and (3.12). Following the procedure of the paper of Abdullah and Shah [15] it can be easily shown that there exists functions wk : 1 ≤ k ≤ q N − 1 satisfying conditions (3.15) and (3.16) if and only if the function M0 is of the form    2    wˆ 0 (ξ ) 2  wˆ 0 ξ + u(N )  M0 (ξ ) =  √  +  √  ,   q N q N

M. Y. Bhat

and satisfies the following identity   M0 ξ + p2 = M0 (ξ ). Theorem 3.8 For each i ∈ {0, 1, . . . , q N − 1} and  ∈ N let f ,i ∈ l 2 (). Then the system   G = T(q N ) λ f ,i : λ ∈ , 0 ≤ i ≤ q N − 1 is an orthonormal in l 2 () if and only if for 0 ≤ i, j ≤ q N − 1, the following conditions are satisfied: Tij (ξ ) =

(q N ) −1  

 fˆ,i ξ +

s=0

 + fˆ,i ξ +

u(s) (p−1 N )



u(s) + u(N ) (p−1 N )

 fˆ, j ξ +



u(s) (p−1 N )

 fˆ, j ξ +



u(s) + u(N ) (p−1 N )



= q(q N ) δi, j ,

(3.20)       r u(s) u(s) pu(s) χ fˆ,i ξ + −1  fˆ, j ξ + −1  N (p N ) (p N ) s=0     u(s) u(s) ˆ ˆ + f ,i ξ + −1  + u(N ) f , j ξ + −1  + u(N ) = 0. (p N ) (p N )

(q N ) −1 

(3.21)

Proof For λ, σ ∈  and 0 ≤ i, j ≤ q N − 1, the orthonormality of the system G in l 2 () is equivalent to 

 T(q N ) λ f ,i , T(q N ) σ f , j = δλ,σ δi, j .

By setting     h i j (ξ ) = fˆ,i (ξ ) fˆ, j (ξ ) + fˆ,i ξ + u(N ) fˆ, j ξ + u(N ) , and taking λ = u(m) and σ = u(n), where m, n ∈ N0 , we have δλ,σ δi, j =

pD

=

    h i j (ξ )χ p−1 (p−1 N ) u(m) − u(n) ξ dξ ⎧ ) −1 ⎨(q N

(p/(q N ) )D ⎩

s=0

 hi j ξ +

⎫ ⎬     u(s) χ (p−1 N ) u q(m − n) ξ dξ. −1  (p N ) ⎭

Therefore following the procedure of the proof of the Theorem 3.1, we get the desired result.  

Nonuniform Discrete Wavelets on Local Fields of Positive…

4 Nonuniform Discrete Wavelet System for l 2 ()   Theorem 4.1 Let the system Tq N λ wi : wi ∈ l 2 (); λ ∈ ; 0 ≤ i ≤ q N − 1 be orthonormal in l 2 (), where wi ∈ l 2 () such that (3.19) holds. Also, for  ∈ N and f (−1),i ∈ l 2 (), let the system 

 Tq N λ f (−1),i : λ ∈ , 0 ≤ i ≤ q N − 1

(4.1)

be orthonormal in l 2 (). Consider the following relation   fˆ,i (ξ ) = fˆ(−1),i (ξ )wˆ i (p−1 N )−1 ξ ,

(4.2)

where fˆ0,0 (ξ ) = 1, a.e. Then the system 

 T(q N ) λ f ,i : λ ∈ ; 0 ≤ i ≤ q N − 1

is orthonormal in l 2 (). Proof For  = 1, the result is trivial. Assume  ∈ N − {1} and 0 ≤ i, j ≤ q N − 1. For the proof, it is sufficient to show the equations (3.20) and (3.21). We observe that fˆ,i ∈ L 2 ( ) as & &2   2 ˆ &ˆ &  f (−1),i (ξ )wˆ i (p−1 N )−1 ξ  dξ & f ,i & = 2



&2   2 & & & ≤ supξ wˆ i (p−1 N )−1 ξ  & fˆ(−1),i & 2

&2 & & & = q N & fˆ(−1),i & . 2

For Eq. (3.20), we have

Ti j (ξ )

=

(q  N ) −1



 ˆ f ,i ξ +

s=0

 ˆ + f ,i ξ +

u(s) −1 (p N )



 ˆ f , j ξ +

u(s) + u(N ) (p−1 N )



u(s) −1 (p N )

 ˆ f , j ξ +



u(s) + u(N ) (p−1 N )

  2   u(n) u(m)   ˆ =  f (−1),0 ξ + (p−1 N )−1 + (p−1 N )  m=0 n=0   2    u(n) u(m) +  fˆ(−1),0 ξ + −1 −1 + −1  + u(N )  (p N ) (p N ) q N −1 (q  N ) −1



M. Y. Bhat





 u(m) × wˆ i (p N ) ξ + pu(n) + −1 p N   u(m) × wˆ i (p−1 N )−1 ξ + pu(n) + −1 , p N −1

−1

  where we have the fact wˆ i ξ + u(N ) = wˆ i (ξ ) and Eq. (4.2). Furthermore, for 0 ≤ m ≤ q N − 1 and 0 ≤ n ≤ (q N )−1 − 1, define   2   u(n) u(m)  ˆ Fm,n (ξ ) =  f (−1),0 ξ + −1 −1 + −1   (p N ) (p N )  2    u(m) u(n) +  fˆ(−1),0 ξ + −1 −1 + −1  + u(N )  . (p N ) (p N ) Then, we have

Ti j (ξ )

=



  u(m) Fm,n (ξ ) wˆ i (p−1 N )−1 ξ + pu(n) + −1 p N m=0 n=0    u(m) × wˆ i (p−1 N )−1 ξ + pu(n) + −1 . p N

q N −1 (q  N ) −1

For each 0 ≤ i, j ≤ q N − 1, we have by Eq. (3.19) Ti j (ξ )

=

q N −1 m=0

  u(m) −1 −1 wˆ i0 (p N ) ξ + −1 p N

  (q  N ) −1 u(m) Fm,n (ξ ) × wˆ i0 (p−1 N )−1 ξ + −1 p N n=0

+

q N −1 m=0

  u(m) wˆ i1 (p−1 N )−1 ξ + −1 p N

  (q  N ) −1 u(m) −1 −1 Fm,n (ξ ) × wˆ i1 (p N ) ξ + −1 p N n=0       q N −1 r u(m) u(m) (p−1 N )−1 + −1 wˆ i1 (p−1 N )−1 ξ + −1 + χ N p N p N m=0 ⎫ ⎧   ⎨(q  N ) −1   ⎬  u(m) r χ × wˆ j0 (p−1 N )−1 ξ + −1 pu(n) Fm,n (ξ ) ⎭ ⎩ p N N n=0

Nonuniform Discrete Wavelets on Local Fields of Positive…

      r u(m) u(m) (p−1 N )−1 + −1 wˆ i0 (p−1 N )−1 ξ + −1 + χ N p N p N m=0 ⎫ ⎧   ⎨(q  N ) −1   ⎬  u(m) r χ × wˆ j1 (p−1 N )−1 ξ + −1 pu(n) Fm,n (ξ ) . ⎭ ⎩ p N N q N −1

n=0

By Theorem 3.8 and the orthonormality property of the system (4.1), we obtain (q  N ) −1

Fm,n (ξ ) = q(q N )

−1

n=0

and

(q  N ) −1 

χ

n=0

r N

pu(n)



Fm,n (ξ ) = 0,

which in turn imply that

Ti j (ξ )

  u(m) −1 −1 wˆ i0 (p N ) ξ + −1 = q(q N ) p N m=0   u(m) × wˆ i0 (p−1 N )−1 ξ + −1 p N     u(m) u(m) −1 −1 −1 −1 + wˆ i1 (p N ) ξ + −1 wˆ i1 (p N ) ξ + −1 . p N p N −1

q N −1

Note that Ti j (ξ ) = q(q N )−1 (q N δi j ) = q(q N ) δi j , as the system (3.13) is orthonormal in l 2 (). This proves the Eq. (3.20). Similarly, we can prove Eq. (3.21). This completes the proof.   Next we invoke Theorem 4.1 to provide orthogonal splitting of the subspaces V j s. Theorem 4.2 With the assumptions of Theorem 4.1, let us define the subsets V−1 , V and W of V0 = l 2 () by   V−1 = span T(q N )−1 λ f (−1),0 : λ ∈  ,   V = span T(q N ) λ f ,0 : λ ∈  ,   W = span T(q N ) λ f ,i : λ ∈ , 1 ≤ i ≤ q N − 1 . Then V ⊕ W = V−1 .

M. Y. Bhat

Proof For  ∈ N and 0 ≤ i ≤ q N − 1, we can write      wˆ i (p−1 N )−1 ξ = wi (ν)χ (p−1 N )−1 νξ ν∈

=



      wi σ − p−1 N λ χ (p−1 N )−1 σ − p−1 N λ ξ ,

σ ∈

for some λ ∈  and then we have     χ (p−1 N ) λξ wˆ i (p−1 N )−1 ξ fˆ(−1),0 (ξ )      = wi σ − p−1 N λ χ (p−1 N )−1 σ ξ fˆ(−1),0 (ξ ), σ ∈

or        χ (p−1 N ) λξ fˆ,i (ξ ) = wi σ − p−1 N λ χ (p−1 N )−1 σ ξ fˆ(−1),0 (ξ ), σ ∈

or 

T(q N ) λ f ,i

∧

(ξ ) =

 σ ∈

 ∧  wi σ − p−1 N λ T(q N )−1 σ f −1,i (ξ ),

or T(q N ) λ f ,i (ξ ) =

 σ ∈

  wi σ − p−1 N λ T(q N )−1 σ f −1,i (ξ ).

Hence V and W are subspaces of V−1 .   Using the facts that V is orthogonal to W as the systems T(q N ) λ f ,0 : λ ∈    and T(q N ) λ f ,i : λ ∈ , 1 ≤ i ≤ q N − 1 are orthogonal and V ⊕ W ⊂ V−1 , so to complete the proof, it suffices to show that V−1 ⊂ V ⊕ W . Thus, we have T(q N )−1 λ f (−1),0 (ξ )    = wi σ − p−1 N λ T(q N ) σ f −1,0 (ξ ) σ ∈

=

σ ∈

=

⎩

i=0 ν∈

      −1 wi σ − p N λ , Tq N ν wi Tq N ν wi (σ )T(q N ) σ f −1,0 (ξ )

q N −1   i=0 ν∈

=

⎫ ⎬    wi σ − p−1 N λ , Tq N ν wi Tq N ν wi (σ ) T(q N ) σ f −1,0 (ξ ) ⎭

⎧ N −1    ⎨q

σ ∈

q N −1   i=0 ν∈

   wi σ − p−1 N λ , Tq N ν wi T(q N ) ν f ,i (ξ )

Nonuniform Discrete Wavelets on Local Fields of Positive…

=

 ν∈

+

   wi σ − p−1 N λ , Tq N ν wi T(q N ) ν f ,0 (ξ )

q N −1   i=1 ν∈

   wi σ − p−1 N λ , Tq N ν wi T(q N ) ν f ,i (ξ ),

which shows that V−1 ⊂ V ⊕ W . This completes the proof.

 

Theorem 4.3 For each , 1 ≤  ≤ s, and i, 0 ≤ i ≤ q N − 1, let w,i ∈ l 2 () such that wˆ ,i (ξ ) = w,i0 (ξ ) + χ

r  ξ w,i1 (ξ ), N

(4.3)

for some L 2 , p-periodic functions w,i0 and w,i1 . For each , assume that the matrix M (ξ ) is unitary, with its entries defined by   ⎧ u(s) ⎪ ⎪ ξ + ; 0 ≤ s ≤ q N − 1; 0 ≤ t ≤ q N − 1, w ⎪ ,t0 ⎪ p−1 N ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ u(s − q N ) ⎪ ⎪ w,t1 ξ + ; q N ≤ s ≤ q 2 N − 1; 0 ≤ t ≤ q N − 1, ⎪ ⎪ p−1 N ⎪ ⎪ ⎪ ⎪ ⎪   ⎪  ⎪ r ⎪ u(s − q N ) 1 ⎨χ pu(s) w,(t−q N )0 ξ + ; 0 ≤ t ≤ q N − 1; M,st (ξ ) = √ N p−1 N q N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ q N ≤ s ≤ q 2 N − 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪   ⎪  ⎪ r u(s − q N ) ⎪ ⎪ pu(s) w ; q N ≤ s ≤ q 2 N − 1; ξ + χ ⎪ ,(t−q N )1 ⎪ ⎪ N p−1 N ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ q N ≤ t ≤ q 2 N − 1.

For given  and i, define f ,i as follows  fˆ,i (ξ ) = fˆ−1,i (ξ )wˆ ,i (p−1 N )−1 ξ ), f or 2 ≤  ≤ s with f 1,i = w1,i and fˆ0,0 = 1, a.e. Then ' V0 = l () = Vs ⊕ 2

s 

( Wm ,

(4.4)

m=1

where W j = V j+1  V j , j ∈ Z and the set   H = T(q N ) λ f ,i , T(q N )s λ f s,0 : 1 ≤  ≤ s, 1 ≤ i ≤ q N − 1, λ ∈ 

(4.5)

M. Y. Bhat

is an orthonormal basis for l 2 (). We call it as the sth-stage nonuniform discrete wavelet system for l 2 (). Proof From Theorem 4.1, it follows that for each , 1 ≤  ≤ s, the system 

 T(q N ) λ f ,i : 1 ≤ i ≤ q N − 1, λ ∈ 

(4.6)

is orthonormal in l 2 (). Therefore, the system 

 T(q N )s λ f s,0 : λ ∈ 

and the system defined by (4.5) are orthonormal for each . Further, using Theorem 4.2, it follows that for each , 1 ≤  ≤ s, V ⊂ V−1 and V is orthogonal to W in V−1 . This means that W is orthogonal to W−1 for each . Therefore, the system H defined by (4.5) is orthonormal in l 2 (). Since V ⊕ W = V−1 , so we can write   V0 = V1 ⊕ W1 = V2 ⊕ W1 ⊕ W2 = · · · = Vs ⊕ ⊕sm=1 Wm . As (4.4) holds, the system (4.5) is orthonormal in l 2 (). This completes the proof.   Theorem 4.4 Assume that the conditions of the Theorem 4.3 hold. Moreover for each  ∈ N0 , define V by   V0 = l 2 (), V = span T(q N ) λ f ,0 : λ ∈  .  ∞ ∞ 2 2 Then ∞ =0 V = l (). Further, if =0 V = {0}, then l () = ⊕m=1 Wm , where W j = V j+1 ⊕ V j , j ∈ Z, and for  ∈ N the system (4.6) is an orthonormal basis for l 2 (). Proof Since, for each  ∈ N, V ⊂ V−1 and V0 = l 2 (), it follows that ∞ )

V = l 2 ().

=0

Using the fact V ⊕ W = V−1 , we have   V0 = V1 ⊕ W1 = V2 ⊕ W1 ⊕ W2 = · · · = Vs ⊕ ⊕sm=1 Wn . To show V0 = ⊕∞ m=1 Wm , it is sufficient to show that the orthogonal complement of W in V is {0}. For this, suppose f ∈ V0 is orthogonal to ⊕∞ ⊕∞ m 0 m=1 m=1 Wm . Then f is orthogonal to each Wm for m ∈ N. Thismeans that f is a member of each Vm as Wm is orthogonal to Vm . Therefore, f ∈ ∞ =0 V = {0}, which means that f = 0, a.e. This completes the proof.  

Nonuniform Discrete Wavelets on Local Fields of Positive…

Definition 4.5 If the system 

 T(q N ) λ f ,i : f ,i ∈ l 2 (),  ∈ N, 1 ≤ i ≤ q N − 1, λ ∈ 

is orthonormal in l 2 (), then it is called nonuniform discrete wavelet system for l 2 ().

5 Relationship with NUMRA Wavelets of L 2 (K ) In this section, we provide a connection between first-stage nonuniform discrete wavelet system for l 2 () and a system of NUMRA wavelets of L 2 (K ).  q N −1 Theorem 5.1 Let ψ =1 be a system of NUMRA wavelets with scaling function ψ0 in L 2 (K ). Then there is a first-stage nonuniform discrete wavelet system for l 2 () associated with a system of NUMRA wavelets of L 2 (K ).  q N −1 Proof Given a system ψ =1 of NUMRA wavelets with scaling function ψ0 in L 2 (K ), we define V j∗ , j ∈ Z as   V j∗ = span D j Tλ ψ0 (x) : λ ∈  ,   where Tλ ψ0 (x) : λ ∈  is an orthonormal basis for V0∗ and the unitary operators Tλ and δ j are defined by Tλ f (x) = f (x − λ),

  D j f (x) = (q N ) j/2 f (p−1 N ) j x , for f ∈ L 2 (K ).

 q N −1  q N −1 Since ψ =1 ⊂ V1∗ , there is w =1 ⊂ l 2 () such that for each , 0 ≤  ≤ q N − 1, 

ψ (x) =

w (λ)DTλ ψ0 (x).

(4.7)

λ∈

Equation (4.7) can be written in the frequency domain as ψˆ  (x) =



w (λ)(DTλ ψ0 (x))∧

λ∈

 = m where m  (ξ ) = can write

√1 qN



p−1 N

λ∈ w (λ)χλ (ξ )

m  (ξ ) = m 0 (ξ ) + χ



ξ

ψˆ 0



ξ p−1 N

 ,

is L 2 locally. Since  = {0, r/N } + Z, we

r  ξ m 1 (ξ ), 0 ≤  ≤ q N − 1, N

(4.8)

M. Y. Bhat

where m 0 and m 1 are locally L 2 , p-periodic functions. Therefore,  we have the following equivalent conditions of the orthonormality of the system Tλ ψ (x) : 0 ≤  ≤  q N − 1, λ ∈ 

(a)

q N −1



t=0

    u(t) u(t) m k0 ξ + −1 m 0 ξ + −1 p N p N

    u(t) u(t) + m 0 ξ + −1 m k0 ξ + −1 = δ,k , p N p N

(b)

q N −1

χ

t=0

r N

pu(t)





    u(t) u(t) m k0 ξ + −1 m 0 ξ + −1 p N p N

    u(t) u(t) + m 0 ξ + −1 m k0 ξ + −1 = 0, p N p N for 0 ≤ , k ≤ q N − 1. From the definition of m  , we see that 1 wˆ  (ξ ) m  (ξ ) = √ qN where wˆ  denotes the Fourier transform in the sense of l 2 (). Now using (3.19), we have 1 1 m 0 (ξ ) = √ w0 (ξ ) and m 1 (ξ ) = √ w1 (ξ ), qN qN where w0 and w1 have same properties as that of m 0 and m 1 . Substituting the values of m 0 and m 1 in (a) and (b), we have for 0 ≤ , k ≤ q N − 1, 

    u(t) u(t) wk0 ξ + −1 w0 ξ + −1 p N p N t=0     u(t) u(t) + w0 ξ + −1 wk0 ξ + −1 = q N δ,k , p N p N      q N −1   u(t) u(t) r pu(t) w0 ξ + −1 (b) χ wk0 ξ + −1 N p N p N t=0     u(t) u(t) + w0 ξ + −1 wk0 ξ + −1 = 0. p N p N (a)

q N −1

These conditions are equivalent to the a.e unitary of the matrix M(ξ ) of order q 2 N × q 2 N with entries as in the Corollary 3.6. Therefore the system (3.13) is an orthonormal

Nonuniform Discrete Wavelets on Local Fields of Positive…

basis of l 2 () and hence is the first-stage nonuniform discrete wavelet system for   l 2 (). By observing that m  and w are closely related for each , following result can be easily proved Theorem 5.2 Let H, defined by (3.13) be a first-stage nonuniform discrete wavelet  q N −1 system for l 2 (). Then there exists a system of NUMRA wavelets ψ =1 with scaling function ψ0 in L 2 (K ) associated with the first-stage nonuniform discrete wavelet system for l 2 (). It is evident from Theorems 5.1 and 5.2 that the nonuniform multiresolution analysis wavelets of L 2 (K ) are connected with the first-stage nonuniform discrete wavelet system of L 2 () and vice-versa. Acknowledgements We are deeply indebted to the referee for his/her valuable suggestions which greatly improved the presentation of this paper.

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