Gabor frame multipliers and Parseval duals on the half real line

Abstract

Recently, Gabor analysis on locally compact abelian (LCA) groups has become the focus of an active research. In practice, the time variable cannot be negative. The half real line [Formula: see text] is an LCA group under multiplication and the usual topology, with the Haar measure [Formula: see text]. This paper addresses Gabor frame multipliers and Parseval duals for [Formula: see text]. We introduce and characterize Gabor frame multipliers and Parseval Gabor frame multipliers based on Zak transform matrices. Our Zak transform matrix is essentially different from the conventional Zibulski–Zeevi matrix. It allows us to define Gabor frame generators by designing suitable matrix-valued functions of finite size. We also prove that an arbitrary Gabor frame [Formula: see text] admits a Parseval dual frame/tight dual frame whenever [Formula: see text] are rational numbers not greater than [Formula: see text].