On Gabor frames generated by B-splines, totally positive functions, and Hermite functions

Abstract

The frame set of a window ϕ∈L2(R) is the subset of all lattice parameters (α,β)∈R+2 such that G(ϕ,α,β)={e2πiβm⋅ϕ(⋅−αk):k,m∈Z} forms a frame for L2(R). In this paper, we investigate the frame set of B-splines, totally positive functions, and Hermite functions. We derive a sufficient condition for Gabor frames using the connection between sampling theory in shift-invariant spaces and Gabor analysis. As a consequence, we obtain a new frame region belonging to the frame set of B-splines and Hermite functions. For a class of functions that includes certain totally positive functions, we prove that for any choice of lattice parameters α,β>0 with αβ<1, there exists a γ>0 depending on αβ such that G(ϕ(γ⋅),α,β) forms a frame for L2(R). Our results give explicit frame bounds.