Let g be a totally positive function of finite type, that is, ?(ξ)=∏~V=1M (1+2πiδ _ν ξ)~(-1)) for δ_ν R, δ_ν ≠0, and M≥2, and let α,β>0. Then the set {e~(2πiβlt)g(t-αk):k,lZ} is a frame for L~2 (R) if and only if αβ<1. This result is a first positive contribution to a conjecture of Daubechies from 1990. Until now, 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 yields an uncountable class of window functions. As a by-product of the proof method, we also derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.
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