Frame set for shifted sinc-function

Abstract

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

13 pages