Group-Frames for Banach Spaces

TL;DR

This paper extends the concept of group-frames from Hilbert spaces to Banach spaces, providing new characterizations and analyzing frames generated by group representations and time-frequency shifts.

Contribution

It introduces the first study of group-frames in Banach spaces, characterizing them via matrices and exploring frames generated by time-frequency shifts.

Findings

Characterization of Banach space group-frames using matrices

Identification of functional-vector pairs generating frames

Derivation of fundamental formulas in Banach space Gabor analysis

Abstract

In the literature, frames generated by unitary representations of groups (known as group-frames) are studied only for Hilbert spaces. We make first study of frames for Banach spaces generated by isometric invertible representations of discrete groups on Banach spaces. These frames are characterized using left regular, right regular, Gram-matrices and group-matrices on classical sequence spaces. A sufficiently large collection of functional-vector pairs using the double commutant of the representation is identified which generate group-frames for Banach spaces. Subsequently, we study Schauder frames generated by time-frequency shift operators on finite dimensional Banach spaces. We derive Moyal formula, fundamental identity of Gabor analysis, Wexler-Raz criterion and Ron-Shen duality in functional form.