Super-wavelets on local fields of positive characteristic

TL;DR

This paper extends the concept of super-wavelets to local fields of positive characteristic, providing characterizations, constructions, and conditions for Parseval frame multiwavelets and super-wavelets in this setting.

Contribution

It develops the theory of super-wavelets on local fields of positive characteristic, including characterizations, group-theoretic constructions, and necessary conditions for decomposability.

Findings

Characterizations of super-wavelets of finite length

Construction of Shannon type multiwavelets

Necessary conditions for decomposable Parseval frame wavelets

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.