Phase retrieval for continuous Gabor frames on locally compact abelian groups

Abstract

In this paper, we study continuous frames from projective representations of locally compact abelian groups of type Ĝ × G . In particular, using the Fourier transform on locally compact abelian groups, we obtain a characterization of maximal spanning vectors. As an application, for G, a compactly generated locally Euclidean locally compact abelian group or a local field with odd residue characteristic, we prove the existence of maximal spanning vectors, hence the phase retrievability, for the associated ( Ĝ × G)-frames.

Spline function generating Gabor systems on the half real line

“…In the last decades, Gabor analysis on $$$ \mathbb{R} $$$ has seen great achievements. Recently, Gabor analysis on locally compact abelian (LCA) groups has interested some mathematicians 1–9 . Let $$$ G $$$ be a second countable LCA group, and denote its dual group by $$$ \hat{G} $$$, which consists of all characters, that is, all continuous homomorphisms from $$$ G $$$ into the torus .…”

“…respectively; the convolution𝑓 * g of two functions 𝑓 , g ∈ L 1 (R) has the form 𝑓 * g(x) = ∫ R 𝑓 (x − 𝑦)g(𝑦)d𝑦, (1.5) and the Fourier transform  con is defined by…”