2011/04/26 by Karlheinz Gröchenig, Joachim Stöckler · 1 citation
Mathematics · #math.FA
paper · pdf · doi:10.1215/00127094-2141944
published as Duke Math. J. 162, no. 6 (2013), 1003-1031
arxiv created 2011/04/26 · arxiv updated 2019/12/19
Let g be a totally positive function of finite type. Then the Gabor set \e2πi βl t g(t-αk), k,l ∈ Z \ is a frame for L2(R), if and only if αβ<1. This result is a first positive contribution to a conjecture of I. Daubechies from 1990. So far the complete characterization of lattice parameters α, β that generate a frame has been known for only six window functions g. Our main result now provides an uncountable class of functions. As a byproduct of the proof method we derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.