On the Gabor frame set for compactly supported continuous functions

TL;DR

This paper extends the known Gabor frame set for certain compactly supported continuous functions, including B-splines and truncated classical functions, by identifying new classes where the frame property holds.

Contribution

It introduces verifiable conditions for functions to belong to this class, extending the Gabor frame set and ensuring dual windows with controlled support.

Findings

The Gabor frame set can be extended for functions supported on an interval of length two.

B-splines $B_N$ for $N \\ge 2$ are included in the new class.

Certain truncated classical functions also belong to this class.

Abstract

We identify a class of continuous compactly supported functions for which the known part of the Gabor frame set can be extended. At least for functions with support on an interval of length two, the curve determining the set touches the known obstructions. Easy verifiable sufficient conditions for a function to belong to the class are derived, and it is shown that the B-splines $B_N, N\ge 2,$ as well as certain "continuous and truncated" versions of several classical functions (e.g., the Gaussian and the two-sided exponential function) belong to the class. The sufficient conditions for the frame property guarantees the existence of a dual window with a prescribed size of the support.