Gabor frame operator for the Cauchy kernel

TL;DR

This paper analyzes the Gabor frame operator generated by the Cauchy kernel, providing bounds, explicit formulas for the dual window, and insights into the frame's properties across different lattice parameters.

Contribution

It offers explicit expressions for the Gabor frame operator and dual window for the Cauchy kernel, expanding understanding of Gabor frames in this context.

Findings

Derived frame bounds estimates for Gabor frames with Cauchy kernel.

Explicit formula for the canonical dual window for all lattice parameters.

Analyzed the Gabor frame operator's properties across different lattice configurations.

Abstract

We obtain frame bounds estimates and the Gabor frame operator $S=S^{\alpha,\beta}$ for Gabor frames generated by the Cauchy kernel. In addition we find the explicit expression for the canonical dual window for all values of the lattice parameters $\alpha,\beta$, $\alpha\beta\leq 1$.