Step Refinable Functions and Orthogonal MRA on Vilenkin Groups

Non-Haar MRA on local fields of positive characteristic

“…Let ϕ ∈ D M (K −N ) and let (ϕ(x−h)) h∈H 0 be an orthonormal system. V n ⊂ V n+1 if and only if the function ϕ(x) is a solution of refinement equation(5.2).The proof repeats the proof of Theorem 4.2 in[10].…”

“…The proof repeats the proof of Lemma 4.1 in [10]. The refinement equation (5.2) may be written in the form…”

Nonuniform wavelet packets on local fields of positive characteristic

“…In recent years there has been a considerable interest in the problem of constructing wavelet bases on various groups, namely, Cantor dyadic groups [10], locally compact Abelian groups [7], p-adic fields [9] and Vilenkin groups [11]. Recently, R. L. Benedetto and J. J. Benedetto [2] developed a wavelet theory for local fields and related groups.…”

How to construct wavelets on local fields of positive characteristic

“…Construction of non-Haar wavelets is the a basic problem in this theory. The problem of constructing orthogonal MRA on the field F (1) is studied in detail in the works [6,7,8,12,16,17]. S.F.Lukomskii, A.M.Vodolazov [15,18] considered local field F (s) as a vector space over the finite field GF (p s ) and constructed non-Haar wavelets.…”