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On a necessary condition for B-spline Gabor frames

Abstract:
In a previous note K. Gr$\bf \rm{ \ddot{o}}$chenig et al. prove that if $g$ is a continuous function with compact support such that the translates of $g$ form a partition of unity, then $g$ cannot generate a Gabor frame for integer values of the frequency shift parameter $b$ greater than 1. We give a simpler proof of this result which applies also to windows $g$ which are neither continuous nor with compact support. Our proof is based on a necessary condition for Gabor frames due to C. E. Heil and D. F. Walnut.

Keywords:
Gabor frame, partition of unity, dual window, Ron-Shen condition, B-spline

MSC:
42

Altre info:
Il lavoro è stato accettato dalla rivista Ricerche di Matematica, uscirà nei prossimi mesi.