Frame sets for generalized $B$-splines

Abstract

The frame set of a function $g\in L^2(\mathbb{R})$ is the subset of all parameters $(a,b)\in {\mathbb{R}}_{+}^2$ for which the time-frequency shifts of $g$ along $a\mathbb{Z}\times b\mathbb{Z}$ form a Gabor frame for $L^2(\mathbb{R}).$ In this paper, we investigate the frame set of a class of functions that we call \emph{generalized $B-$splines} and which includes the $B-$splines. In particular, we add many new points to the frame sets of these functions. In the process, we generalize and unify some recent results on the frame sets for this class of functions.

23 pages, 4 figures. This is version 2, some typos where corrected, figures of dual functions are given and it is proved that the duals we constructed are discontinuous

References in corpus (1)

• Sampling Theorems for Shift-invariant Spaces, Gabor Frames, and Totally Positive Functions