Wavelet packets and wavelet frame packets on local fields

TL;DR

This paper extends the theory of wavelet packets and frames to local fields of positive characteristic, providing new mathematical tools for analysis in these fields.

Contribution

It generalizes multiresolution analysis and wavelet concepts to local fields, introduces a splitting lemma, and constructs wavelet packets and frame packets in this setting.

Findings

Wavelet packets form orthonormal bases via translations.

Construction of wavelet frame packets in local fields.

Generalization of splitting lemma for frames.

Abstract

Using a prime element of a local field K of positive characteristic p, the concepts of multiresolution analysis (MRA) and wavelet can be generalized to such a field. We prove a version of the splitting lemma for this setup and using this lemma we have constructed the wavelet packets associated with such MRAs. We show that these wavelet packets generate an orthonormal basis by translations only. We also prove an analogue of splitting lemma for frames and construct the wavelet frame packets in this setting.