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We provide a characterization of wavelets on local fields of positive characteristic based on results on affine and quasi affine frames. This result generalizes the characterization of wavelets on Euclidean spaces by means of two basic equations. We also give another characterization of wavelets. Further, all wavelets which are associated with a multiresolution analysis on a such a local field are also characterized. 1 2 BISWARANJAN BEHERA AND QAISER JAHAN Preliminaries on local fieldsLet K be a field and a topological space. Then K is called a locally compact field or 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. So by a local field, we mean a field K which is locally compact, nondiscrete and totally disconnected.We use the notation of the book by Taibleson [20]. Proofs of all the results stated in this section can be found in the books [20] and [18].Let K be a local field. Since K + is a locally compact abelian group, we choose a Haar measure dx for K + . If α = 0, α ∈ K, then d(αx) is also a Haar measure. Let d(αx) = |α|dx. We call |α| the absolute value or valuation of α. We also let |0| = 0.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; (c) |x + y| ≤ max{|x|, |y|} for all x, y ∈ K. Property (c) is called the ultrametric inequality. It follows that |x + y| = max{|x|, |y|}if |x| = |y|.The set D = {x ∈ K : |x| ≤ 1} is called the ring of integers in K. It is the unique maximal compact subring of K. Define P = {x ∈ K : |x| < 1}. The set P is called the prime ideal in K. The prime ideal in K is the unique maximal ideal in D. It is principal and prime.Since K is totally disconnected, the set of values |x| as x varies over K is a discrete set of the form {s k : k ∈ Z} ∪ {0} for some s > 0. Hence, there is an element of P of maximal absolute value. Let p be a fixed element of maximum absolute value in P. Such an element is called a prime element of K. Note that as an ideal in D, P = p = pD.It can be proved that D is compact and open. Hence, P is compact and open. Therefore, the residue space D/P is isomorphic to a finite field GF (q), where q = p c for some prime p and c ∈ N. For a proof of this fact we refer to [20].For a measurable subset E of K, let |E| = K χ E (x)dx, where χ E is the characteristic function of E and dx is the Haar measure of K normalized so that |D| = 1. Then, it is easy to see that |P| = q −1 and |p| = q −1 (see [20]). It follows that if x = 0, and x ∈ K, then |x| = q k for some k ∈ Z.Let D * = D \ P = {x ∈ K : |x| = 1}. D * is the group of units in K * . If x = 0, we can write x = p k x ′ , with x ′ ∈ D * .Recall that D/P ∼ = GF (q). Let U = {a i : i = 0, 1, . . . , q − 1} be any fixed full set of coset representatives of P in D. L...

Using a prime element of a local field K of positive characteristic p, the concepts of multiresolution analysis (MRA) and wavelet can be generalized to such a field. We prove a version of the splitting lemma for this setup and using this lemma we have constructed the wavelet packets associated with such MRAs. We show that these wavelet packets generate an orthonormal basis by translations only.We also prove an analogue of splitting lemma for frames and construct the wavelet frame packets in this setting.[15] gave a definition of MRA on a local field of positive characteristic p and, similar to R n , have constructed the wavelets from an MRA.In this article we construct the wavelet packets associated with such an MRA. We also generalize the wavelet frame packets to this setup. First of all, we will discuss about wavelet packets very briefly.Let {V j : j ∈ Z} be an MRA of L 2 (R) with scaling function ϕ and wavelet ψ.Let W j be the corresponding wavelet subspaces: W j = span{ψ jk : k ∈ Z}. In the construction of a wavelet from an MRA, essentially the space V 1 is split into two orthogonal components V 0 and W 0 . Note that V 1 is the closure of the linear span of the functions {2 1/2 ϕ(2·−k) : k ∈ Z}, whereas V 0 and W 0 are respectively the closure of the span of {ϕ(·−k) : k ∈ Z} and {ψ(·−k) : k ∈ Z}. Since ϕ(2·−k) = ϕ (2(· − 2 −1 k)), we see that the above procedure splits the half integer translates of a function into integer translates of two functions.In a similar way, we can split W j , which is the span of {ψ(2 j · −k) : k ∈ Z} = {ψ (2 j (· − 2 −j k)) : k ∈ Z}, to get two functions whose 2 −(j−1) k translates will span the same space W j . Repeating the splitting procedure j times, we get 2 j functions whose integer translates alone span the space W j . If we apply this to each W j , then the resulting basis of L 2 (R) will consist of integer translates of a countable number of functions (instead of all dilations and translations of the wavelet ψ). This basis is called the "wavelet packet basis". The concept of wavelet packets was introduced by Coifman, Meyer and Wickerhauser [9,10]. For a nice exposition of wavelet packets of L 2 (R) with dilation 2, we refer to [14].The concept of wavelet packet was subsequently generalized to R n by taking tensor products [8]. The non-tensor product versions are due to Shen [25] for dyadic dilation, and Behera [2] for MRAs with a general dilation matrix and several scaling functions.Other notable generalizations are the biorthogonal wavelet packets [7], non-orthogonal

Abstract. The concept of quasi-affine frame in Euclidean spaces was introduced to obtain translation invariance of the discrete wavelet transform. We extend this concept to a local field K of positive characteristic. We show that the affine system generated by a finite number of functions is an affine frame if and only the corresponding quasi-affine system is a quasi-affine frame. In such a case the exact frame bounds are equal. This result is obtained by using the properties of an operator associated with two such affine systems. We characterize the translation invariance of such an operator. A related concept is that of co-affine system. We show that there do not exist any co-affine frame in L 2 (K).