Super‐wavelets on local fields of positive characteristic

Abstract

The concept of super‐wavelet was introduced by Balan, and Han and Larson over the field of real numbers which has many applications not only in engineering branches but also in different areas of mathematics. To develop this notion on local fields having positive characteristic we obtain characterizations of super‐wavelets of finite length as well as Parseval frame multiwavelet sets of finite order in this setup. Using the group theoretical approach based on coset representatives, further we establish Shannon type multiwavelet in this perspective while providing examples of Parseval frame (multi)wavelets and (Parseval frame) super‐wavelets. In addition, we obtain necessary conditions for decomposable and extendable Parseval frame wavelets associated to Parseval frame super‐wavelets.

Characterization of Multiframelet Set on Local Fields of Positive Characteristic

“…In this section we discuss different properties of super-wavelet and associated set. Now considering ψ j = 1 W j and ϕ j = 1 V j for some measurable W j , V j ⊂ K, the equation ( 5) changes to Shukla and Maury [6] gave following necessary condition for a Parseval framelet to be decomposable. dξ ≥ m q − 1 q .…”

“…Motivated from the recent works of Behara and Jahan [3,5], Shukla and Maury [6], our aim is to develop the theory of multiwavelet set, scaling set, Parseval scaling set, Parseval framelet set in local field of positive characteristic(LFPC).…”

On the nonhomogeneous wavelet bi-frames for reducing subspaces of Hs(K)

“…Although their topology is similar, their MRA (multiresolution analysis) and wavelet theories are quite different. The construction of wavelets, wavelet frames, MRA and other related works on local fields of positive characteristics (LFPC)have been studied by Benedetto, Behera and Jahan, Ahmad and Shah, Jiang, Li and Ji, Shukla and Mittal, Bhat, etc in the series of works [2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,25,26,27]. However the research on local fields is still in its infancy.…”