Characterization of wavelets and MRA wavelets on local fields of positive characteristic

Abstract

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...

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