# Tight Wavelet Frame Sets in Finite Vector Spaces

## TL;DR

This paper characterizes conditions under which subsets of finite vector spaces generate tight wavelet frames via inverse Fourier transforms, and provides a constructive method for such sets in certain finite field settings.

## Contribution

It establishes sufficient conditions for tight wavelet frame sets in finite vector spaces and introduces a new constructive approach for their generation.

## Key findings

- Conditions based on tilings for wavelet frame sets
- Construction method for tight wavelet frames in finite fields
- Extension of prior finite field wavelet theory

## Abstract

Let $q\geq 2$ be an integer, and $\Bbb F_q^d$, $d\geq 1$, be the vector space over the cyclic space $\Bbb F_q$. The purpose of this paper is two-fold. First, we obtain sufficient conditions on $E \subset \Bbb F_q^d$ such that the inverse Fourier transform of $1_E$ generates a tight wavelet frame in $L^2(\Bbb F_q^d)$. We call these sets (tight) wavelet frame sets. The conditions are given in terms of multiplicative and translational tilings, which is analogous with Theorem 1.1 ([20]) by Wang in the setting of finite fields. In the second part of the paper, we exhibit a constructive method for obtaining tight wavelet frame sets in $\Bbb F_q^d$, $d\geq 2$, $q$ an odd prime and $q\equiv 3$ (mod 4).