Gabor Frames and Totally Positive Functions

By Karlheinz Gröchenig, J. Stockler. 2011

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Let g be a totally positive function of finite type, i.e., ĝ(ξ) = ∏M ν=1(1 + 2πiδνξ) −1 for δν ∈ R and M ≥ 2. Then the set {eg(t − αk) : k, l ∈ Z} is a frame for L(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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/822f8db8e2e1f29708aff8cf429fa537033fd0a2