Gabor frames for rational functions

Abstract

We study the frame properties of the Gabor systems G(g;α,β):={e2πiβmxg(x-αn)}m,n∈Z.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathfrak {G}}(g;\\alpha ,\\beta ):=\\{e^{2\\pi i \\beta m x}g(x-\\alpha n)\\}_{m,n\\in {\\mathbb {Z}}}. \\end{aligned}$$\\end{document}In particular, we prove that for Herglotz windows g such systems always form a frame for L^2({mathbb {R}}) if alpha ,beta >0, alpha beta le 1. For general rational windows gin L^2({mathbb {R}}) we prove that {mathfrak {G}}(g;alpha ,beta ) is a frame for L^2({mathbb {R}}) if 0<alpha ,beta , alpha beta <1, alpha beta not in {mathbb {Q}} and {hat{g}}(xi )ne 0, xi >0, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of L^2({mathbb {R}}).