Tight wavelet frame sets in finite vector spaces

Abstract

Let q ≥ 2 be an integer, and F q d , d ≥ 1 , be the vector space over the cyclic space F q . The purpose of this paper is two-fold. First, we obtain sufficient conditions on E ⊂ F q d such that the inverse Fourier transform of 1 E generates a tight wavelet frame in L 2 ( 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 F q d , d ≥ 2 , q an odd prime and q ≡ 3 (mod 4).