Frame Multiresolution Analysis on Local Fields of Positive Characteristic

We present a notion of framemultiresolution analysis on local fields of positive characteristic based on the theory of shift-invariant spaces. In contrast to the standard setting, the associated subspace V 0 of L2(K) has a frame, a collection of translates of the scaling function φ of the form {φ(⋅ − u(k)) : k ∈ N 0 }, where N 0 is the set of nonnegative integers. We investigate certain properties of multiresolution subspaces which provides the quantitative criteria for the construction of frame multiresolution analysis (FMRA) on local fields of positive characteristic. Finally, we provide a characterization of wavelet frames associated with FMRA on local fieldK of positive characteristic using the shift-invariant space theory.


Introduction
Multiresolution analysis is considered as the heart of wavelet theory.The concept of multiresolution analysis provides a natural framework for understanding and constructing discrete wavelet systems.A multiresolution analysis is an increasing family of closed subspaces {  :  ∈ Z} of  2 (R) such that ⋂ ∈Z   = {0} and ⋃ ∈Z   is dense in  2 (R) which satisfies  ∈   if and only if (2⋅) ∈  +1 .Furthermore, there exists an element  ∈  0 such that the collection of integer translates of function , {(⋅ − ) :  ∈ Z}, represents a complete orthonormal system for  0 .The function  is called the scaling function or the father wavelet.The concept of multiresolution analysis has been extended in various ways in recent years.These concepts are generalized to  2 (R  ), to lattices different from Z  , 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  ≥ 2 or by an expansive matrix  ∈   (R) as long as  ⊂ Z  (see [1,2]).
On the other hand, this elegant tool for the construction of wavelet bases has been extensively studied by several authors on the various spaces, namely, Cantor dyadic groups [3], locally compact Abelian groups [4], -adic fields [5], zero-dimensional groups [6], and Vilenkin groups [7].Recently, R. L. Benedetto and J. J. Benedetto [8] 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.The 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 -adic field Q  whereas local fields of positive characteristic are the Cantor dyadic group and the Vilenkin -groups.The structures and metrics of the local fields of zero and positive characteristic are similar, but their wavelet and MRA theory are quite different.The concept of multiresolution analysis on a local field  of positive characteristic was introduced by Jiang et al. [9].They pointed out a method for constructing orthogonal wavelets on local field  with a constant generating sequence.Subsequently, tight wavelet frames on local fields of positive characteristic were constructed by Shah and Debnath [10] using extension principles.As far as the characterization of wavelets on local fields is concerned, Behera and Jahan [11] have given the characterization of all wavelets associated with multiresolution analysis on local field  based on results on affine and quasiaffine frames.Recently, Shah and Abdullah [12] have introduced the notion of nonuniform multiresolution analysis on local field  of positive characteristic and 2 Journal of Operators obtained the necessary and sufficient condition for a function  to generate a nonuniform multiresolution analysis on local fields.More results in this direction can also be found in [13,14] and the references therein.
Since the use of multiresolution analysis has proven to be a very efficient tool in wavelet theory mainly because of its simplicity, it is of interest to try to generalize this notion as much as possible while preserving its connection with wavelet analysis.In this connection, Benedetto and Li [15] considered the dyadic semiorthogonal frame multiresolution analysis of  2 (R) with a single scaling function and successfully applied the theory in the analysis of narrow band signals.The characterization of the dyadic semiorthogonal frame multiresolution analysis with a single scaling function admitting a single frame wavelet whose dyadic dilations of the integer translates form a frame for  2 (R) was obtained independently by Benedetto and Treiber by a direct method [16] and by Kim and Lim by using the theory of shift-invariant spaces [17].Later on, Yu [18] extended the results of Benedetto and Li's theory of FMRA to higher dimensions with arbitrary integral expansive matrix dilations and has established the necessary and sufficient conditions to characterize semiorthogonal multiresolution analysis frames for  2 (R  ).
In this paper, we introduce the notion of frame multiresolution analysis (FMRA) on local field  of positive characteristic by extending the above described methods.We first investigate the properties of multiresolution subspaces, which will provide the quantitative criteria for the construction of FMRA on local fields of positive characteristic.We also show that the scaling property of an FMRA also holds for the wavelet subspaces and that the space  2 () can be decomposed into the orthogonal sum of these wavelet subspaces.Finally, we study the characterization of wavelet frames associated with FMRA on local field  of positive characteristic using the shift-invariant space theory.
The paper is organized as follows.In Section 2, we discuss some preliminary facts about local fields of positive characteristic including the definition of a frame.The notion of frame multiresolution analysis of  2 () is introduced in Section 3 and its quantitative criteria are given by means of Theorem 12.In Section 4, we establish a complete characterization of wavelet frames generated by a finite number of mother wavelets on local field  of positive characteristic.

Preliminaries on Local Fields
Let  be a field and a topological space.Then,  is called a local field if both  + and  * are locally compact Abelian groups, where  + and  * denote the additive and multiplicative groups of , respectively.If  is any field and is endowed with the discrete topology, then  is a local field.Further, if  is connected, then  is either R or C. If  is not connected, then it is totally disconnected.Hence, by a local field, we mean a field  which is locally compact, nondiscrete, and totally disconnected.The -adic fields are examples of local fields.More details are referred to in [19,20].In the rest of this paper, we use N, N 0 , and Z to denote the sets of natural and nonnegative integers and integers, respectively.
Let  be a fixed local field.Then, there is an integer  = p  , where p is a fixed prime element of  and  is a positive integer, and a norm | ⋅ | on  such that for all  ∈  we have || ≥ 0 and for each  ∈  \ {0} we get || =   for some integer .This norm is non-Archimedean; that is, | + | ≤ max{||, ||} for all ,  ∈  and | + | = max{||, ||} whenever || ̸ = ||.Let  be the Haar measure on the locally compact, topological group (, +).This measure is normalized so that ∫ D  = 1, where D = { ∈  : || ≤ 1} is the ring of integers in .Define B = { ∈  : || < 1}.The set B is called the prime ideal in .The prime ideal in  is the unique maximal ideal in D, and hence as a result B is both principal and prime.Therefore, for such an ideal B in D, we have B = ⟨p⟩ = pD.
Let D * = D \ B = { ∈  : || = 1}.Then, it is easy to verify that D * is a group of units in  * and if  ̸ = 0, then we may write  = p    ,   ∈ D * .Moreover, each B  = p  D = { ∈  : || <  − } is a compact subgroup of  + and is known as the fractional ideals of  + (see [19]).Let U = {  } −1 =0 be any fixed full set of coset representatives of B in D; then, every element  ∈  can be expressed uniquely as  = ∑ ∞ ℓ=  ℓ p ℓ with  ℓ ∈ U. Let  be a fixed character on  + that is trivial on D but is nontrivial on B −1 .Therefore,  is constant on cosets of D, implying that if  ∈ B  , then   () = () for  ∈ .Suppose that   is any character on  + ; then, clearly the restriction   |D is also a character on D. Therefore, if {() :  ∈ N 0 } is a complete list of distinct coset representatives of D in  + , then, as it was proved in [20], the set { () :  ∈ N 0 } of distinct characters on D is a complete orthonormal system on D.
The Fourier transform f of a function  ∈  1 () ∩  2 () is defined by It is noted that Furthermore, the properties of Fourier transform on local field are much similar to those on the real line.In particular, Fourier transform is unitary on  2 ().
Let the local field  be of characteristic  > 0 and let  0 ,  1 ,  2 , . . .,  −1 be as above.We define a character  on  as follows: Definition 1.Let H be a separable Hilbert space.A sequence The largest constant  and the smallest constant  satisfying ( 9) are called the upper and the lower frame bound, respectively.A frame is said to be tight if it is possible to choose  =  and a frame is said to be exact if it ceases to be a frame when any one of its elements is removed.An exact frame is also known as a Riesz basis.
The following theorem gives us an elementary characterization of frames.
Theorem 2 (see [15]).A sequence {  :  ∈ N 0 } in a Hilbert space H is a frame for H if and only if there exists a sequence and For  ∈ Z and  ∈ , we define the dilation operator   and the translation operator   as follows: Our study uses the theory of shift-invariant spaces developed in [21,22] where Ŝ() = { f( + ()) ∈  2 (N 0 ) :  ∈ ,  ∈ N 0 }.

Frame Multiresolution Analysis on Local Fields
We first introduce the notion of a frame multiresolution analysis (FMRA) of  2 ().
Definition 3. Let  be a local field of positive characteristic  > 0 and let p be a prime element of .A frame multiresolution analysis of  2 () is a sequence of closed subspaces {  :  ∈ Z} of  2 () satisfying the following properties: (a)   ⊂  +1 for all  ∈ Z; (d) the function  lying in  0 implies that the collection (⋅ − ()) ∈  0 , for all  ∈ N 0 ; (e) the sequence {   := (⋅ − ()) :  ∈ N 0 } is a frame for the subspace  0 .
The function  is known as the scaling function while the subspaces   's are known as approximation spaces or multiresolution subspaces.A frame multiresolution analysis is said to be nonexact and, respectively, exact if the frame for the subspace  0 is nonexact and, respectively, exact.In multiresolution analysis studied in [9], the frame condition is replaced by that of an orthonormal basis or an exact frame.
Next, we establish several properties of multiresolution subspaces that will help in the construction of frame multiresolution analysis on local field  of positive characteristic.The following proposition shows that, for every  ∈ Z, the sequence { , :  ∈ N 0 }, where is a frame for   .
The following theorem is the converse to Theorem 7.
The following two propositions are proved in [23].
Combining all our results so far, we have the following theorem.
Proof.Since  0 is a shift-invariant subspace of  2 (), therefore the system {   :  ∈ N 0 } forms a frame for  0 with frame bounds  and .By Theorem 7 and Lemma 11, it follows that   ⊂  +1 , for every  ∈ Z.Hence, by the definition of   ,  lies in   if and only if (p  .)lies in  0 , while (p −1 ⋅) lies in  +1 if and only if (p +1 ⋅) lies in  0 .Thus,  lies in   if and only if (p −1 ⋅) lies in  +1 .Moreover, by assumption (iii) and Proposition 10, it follows that ⋃ ∈Z   is dense in  2 () and ⋂ ∈Z   = {0}.Thus, the sequence {  :  ∈ Z} satisfies all the conditions to be a frame multiresolution analysis of  2 ().
In order to construct wavelet frames associated with frame multiresolution analysis on local fields  of positive characteristic, we introduce the orthogonal complement subspaces {  :  ∈ Z} of   in  +1 .It is easy to verify that the sequence of subspaces {  :  ∈ Z} also satisfies the scaling property; that is, Theorem 13.Let {  :  ∈ Z} be an increasing sequence of closed subspaces of  2 () such that ⋃ ∈Z   is dense in  2 () and ⋂ ∈Z   = {0}.Let   be the orthogonal complement of   in  +1 , for each  ∈ Z.Then, the subspaces   are pairwise orthogonal and Proof.Assume that  < ; then, ⟨  ,   ⟩ = 0, for any   ∈   as   ⊂  +1 ⊂   .Let   be the orthogonal projection operators from  2 () onto   ; then, lim  → ∞    = , lim  → −∞    = 0, and   = { −    :  ∈  +1 }.Therefore, for any  ∈  2 (), we have Thus, the result of the direct sum follows since  +1 −   is the orthogonal projector from  2 () onto   .

Characterization of Wavelet Frames on Local Fields
In this section, we give the characterization of wavelet frames associated with frame multiresolution analysis on local fields of positive characteristic.First, we will characterize the existence of a function  in  0 , where  0 is the orthogonal complement of  0 in  1 , by virtue of the analysis filters  and , defined as in Section 3.

Theorem 14.
Let  be a periodic function associated with the frame multiresolution analysis {  :  ∈ Z} satisfying the condition (23).Define  0 as the orthogonal complement of  0 in  1 .Let  ∈  1 such that where  is a periodic function in Then, in view of ( 23) and ( 42 Consequently,  = 0, a.e., is the necessary and sufficient condition for (44) to hold for all  ∈ N 0 .(62) Thus, for a given  ∈ Z, the collection { ℓ , :  ∈ N 0 , ℓ = 1, . . ., } constitutes a frame for   with frame bounds  and .
Let  be an arbitrary function in  2 (); then, by Theorem 13   (66) This completes the proof of the theorem.