Gabor frames for rational functions

2021/03/16 by Belov, Yurii, Kulikov, Aleksei, Lyubarskii, Yurii · 5 citations
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2103.08959

Abstract

We study the frame properties of the Gabor systems \mathfrakG(g;α,β):=\e^{2πi βm x}g(x-αn)\_m,n∈ℤ. In particular, we prove that for Herglotz windows g such systems always form a frame for L²(ℝ) if α,β>0, αβ≤1. For general rational windows g∈ L²(ℝ) we prove that \mathfrakG(g;α,β) is a frame for L²(ℝ) if 00, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of L²(ℝ).

Cited by

• Frames for compactly supported functions with irrational density
• Shift-invariant sampling in two-sided small Fock spaces
• Exponential frames and Riesz sequences at the critical density
• Universal frame set for rational functions
• Frame redundancy and Beurling density

Related

• Gabor frames with rational density
• Gabor frames for functions supported on a semi-axis
• Frames for compactly supported functions with irrational density
• On Gabor frames generated by B-splines, totally positive functions, and Hermite functions
• Frame set for Gabor systems with Haar window