Frame set for shifted sinc-function

TL;DR

This paper characterizes the frame set for a shifted sinc-function, showing precise conditions under which the function forms a frame, and extends results to a class of related window functions with specific decay properties.

Contribution

It provides a complete description of the frame set for shifted sinc-functions and extends the characterization to a broader class of window functions with decay conditions.

Findings

Frame set for shifted sinc-function is exactly described by _g=(\u03b1,) :    1,   |b|.

For a class of window functions with specific exponential decay, the frame set is _g=(,) :    1.

The results extend the understanding of Gabor frames for functions with imaginary shifts and particular decay properties.

Abstract

We prove that frame set $\mathcal{F}_g$ for imaginary shift of sinc-function $$g(t)=\frac{\sin\pi b(t-iw)}{t-iw}, \quad b,w\in\mathbb{R}\setminus\{0\}$$ can be described as $\mathcal{F}_g=\{(\alpha,\beta): \alpha\beta\leq 1, \beta\leq|b|\}.$ \\ In addition, we prove that $\mathcal{F}_g=\{(\alpha,\beta): \alpha\beta\leq 1 \}$ for window functions $g$ of the form $\frac{1}{t-iw}(1-\sum\limits_{k=1}^{\infty}a_ke^{2\pi i b_k t})$, such that $\sum_{k\geq 1}|a_k|e^{2\pi|w|b_k}<1$, $wb_k<0$.