Tight Wavelet Frame Sets in Finite Vector Spaces

Let $q\ge 2$ be an integer, and ${\mathbb{F}}_q^d$, $d\ge 1$, be the vector space over the cyclic space ${\mathbb{F}}_q$. The purpose of this paper is two-fold. First, we obtain sufficient conditions on $E\subset {\mathbb{F}}_q^d$ 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 ${\mathbb{F}}_q^d$, $d\ge 2$, $q$ an odd prime and $q\equiv 3$ (mod 4).