# Finite equal norm Parseval Wavelet Frames over Prime Fields

## TL;DR

This paper develops a method to construct finite equal-norm Parseval wavelet frames over prime fields using a scaled DFT approach, providing new characterizations and concrete examples in wavelet analysis.

## Contribution

It introduces a novel scaled DFT technique for finite wavelet frames over prime fields and characterizes subgroups forming frames, advancing wavelet frame theory.

## Key findings

- Constructed finite equal-norm Parseval wavelet frames over prime fields.
- Characterized subgroups of the cyclic multiplicative group that generate wavelet frames.
- Provided concrete examples demonstrating the application of the theoretical results.

## Abstract

In the framework of wave packet analysis, finite wavelet systems are particular classes of finite wave packet systems. In this paper, using a scaling matrix on a permuted version of the discrete Fourier transform (DFT) of system generator, we derive a locally-scaled version of the DFT of system genarator and obtain a finite equal-norm Parseval wavelet frame over prime fields. We also give a characterization of all multiplicative subgroups of the cyclic multiplicative group, for which the associated wavelet systems form frames. Finally, we present some concrete examples as applications of our results.