Constructing Multiresolution Analysis via Wavelet Packets on Sobolev Space in Local Fields

TL;DR

This paper develops a multiresolution analysis framework using wavelet packets on Sobolev spaces over local fields of positive characteristic, introducing fractal functions and proving orthogonality of wavelet packets.

Contribution

It introduces a novel MRA construction on Sobolev spaces over local fields and demonstrates orthogonality of wavelet packets, including fractal examples, using convolution theory.

Findings

Constructed Haar wavelet packets with proven orthogonality.

Developed MRA framework on Sobolev spaces over local fields.

Introduced fractal functions as elements of these spaces.

Abstract

We define Sobolev spaces $H^{\mathfrak{s}}(K_q)$ over a local field $K_q$ of finite characteristic $p>0$, where $q=p^c$ for a prime $p$ and $c\in \mathbb{N}$. This paper introduces novel fractal functions, such as the Weierstrass type and 3-adic Cantor type, as intriguing examples within these spaces and a few others. Employing prime elements, we develop a Multi-Resolution Analysis (MRA) and examine wavelet expansions, focusing on the orthogonality of both basic and fractal wavelet packets at various scales. We utilize convolution theory to construct Haar wavelet packets and demonstrate the orthogonality of all discussed wavelet packets within $H^{\mathfrak{s}}(K_q)$, enhancing the analytical capabilities of these Sobolev spaces.