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

Abstract

The frame set of a function g ∈ L 2 ( R ) is the set of all parameters ( a , b ) ∈ R + 2 for which the collection of time-frequency shifts of g along a Z × b Z form a Gabor frame for L 2 ( 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 uses the compact support of this function to partition a subset of the putative frame set and finds an explicit dual window function in each subregion. Numerical evidence indicates the existence of further regions belonging to the frame set.