Semi-orthogonal Parseval wavelets associated to GMRAs on local fields of positive characteristics

TL;DR

This paper develops a theoretical framework for semi-orthogonal Parseval wavelets linked to GMRAs on local fields of positive characteristic, extending wavelet theory to this mathematical setting.

Contribution

It introduces a characterization of semi-orthogonal Parseval wavelets and related functions on local fields of positive characteristic, expanding wavelet analysis in this context.

Findings

Characterization of semi-orthogonal Parseval wavelets via consistency equations

Conditions for orthonormal (multi)wavelets related to GMRAs

Characterizations of Parseval scaling functions and bandlimited wavelets

Abstract

In this article we establish theory of semi-orthogonal Parseval wavelets associated to generalized multiresolution analysis (GMRA) for the local field of positive characteristics (LFPC). By employing the properties of translation invariant spaces on the core space of GMRA we obtain a characterization of semi-orthogonal Parseval wavelets in terms of consistency equation for LFPC. As a consequence, we obtain a characterization of an orthonormal (multi)wavelet to be associated with an MRA in terms of multiplicity function as well as dimension function of a (multi)wavelet. Further, we provide characterizations of Parseval scaling functions, scaling sets and bandlimited wavelets together with a Shannon type multiwavelet for LFPC.