Gabor frame multipliers and Parseval duals on the half real line (Q6591706)

Inspired by the ongoing active research in Gabor analysis on locally compact abelian (LCA) groups, this paper introduces and investigates Gabor frame multipliers and Parseval Gabor frame multipliers using the matrix representation of the Zak transformation. The study delves into the properties of optimal lower frame bounds for the Parseval dual and tight dual of a given Gabor frame, providing critical insights into their structural characteristics. Additionally, the characterization of Gabor frame multipliers and Parseval Gabor frame multipliers is thoroughly analyzed, revealing their significance in the framework of LCA groups. Furthermore, it is also proved that any arbitrary Gabor frame admits a Parseval dual and tight dual frame when \(\ln a\) and \(\ln b\) are rational numbers not exceeding \(\frac{1}{2}\).