On the frame set of the second-order $B$-spline

TL;DR

This paper investigates the frame set of the second-order B-spline, establishing new regions where it forms a Gabor frame by leveraging its compact support and constructing explicit dual windows.

Contribution

It introduces novel regions of the frame set for the second-order B-spline using a partitioning approach and explicit dual window construction.

Findings

New regions of the frame set are identified.

Explicit dual window functions are constructed.

Numerical evidence suggests additional frame set points.

Abstract

The frame set of a function $g\in L^2(\mathbb{R})$ is the set of all parameters $(a, b)\in \mathbb{R}^2_+$ for which the collection of time-frequency shifts of $g$ along $a\mathbb{Z}\times b\mathbb{Z}$ form a Gabor frame for $L^2(\mathbb{R}).$ Finding the frame set of a given function remains a challenging open problem in time-frequency analysis. In this paper, we establish new regions of the frame set of the second order $B-$spline. Our method relies on the compact support of this function to partition a subset of the putative frame set and find an explicit dual window function in each of the partition regions. Numerical evidence indicates the existence of further points belonging to the frame set.