On Gabor frames and Pisot family substitution tilings

Abstract

Given a function g from the modulation space ℳ¹(ℝ^d) and a Pisot family substitution tiling ϱ on ℝ^2d, we show that there exists a Gabor frame G(g₁, … , g_N ; Λ(Ω)) which is a L²(ℝ^d)-frame and a ℳ^p(ℝ^d)-frame for p ∈ [1,∞], where g_i for i = 1, … , N is a time-frequency translate of g and Λ(Ω) is an aperiodic (Delone) model set based on a cut-and-project scheme for ϱ. Likewise, we prove that the Gabor system G(g₁, … , g_N ; Λ') for any Λ' in the hull X(Λ(Ω)) of Λ(Ω) is also a L²(ℝ^d)-frame and a ℳ^p(ℝ^d)-frame for p ∈ [1,∞].