Gabor like structured frames in separable Hilbert spaces (Q6593691)

This article explores structured frames in separable Hilbert spaces, emphasizing Pseudo \( B \)-Gabor frames. The authors delve into the distinction between traditional Gabor frames and the Pseudo \( B \)-Gabor frames, proposing a framework where invertible operators map from \( L^2(\mathbb{R}) \) into a Hilbert space \( \mathcal{H} \).\N\NThe most interesting contribution appears in Theorem 3.7, which stipulates conditions under which a bounded linear operator \( S \) on \( \mathcal{H} \) qualifies as a Pseudo \( B \)-Gabor-like frame operator. The work defines \( S \) through the relation \( S = T T^\ast \), where \( T = B A \) and \( A \) commute with specific translation and modulation operators.\N\NThe significance of the article lies in extending the framework of frame operators beyond classical Gabor systems. The discussion on the conditions for the existence of Parseval Pseudo \( B \)-Gabor frames and their properties, such as positivity and commutativity, adds depth to the literature. This study offers insights into generating new types of frames in \( \mathcal{H} \), maintaining a structure analogous to Gabor frames in \( L^2(\mathbb{R}) \).