Gabor frames for rational functions

Abstract

AbstractWe study the frame properties of the Gabor systems $$\begin{aligned} {\mathfrak {G}}(g;\alpha ,\beta ):=\{e^{2\pi i \beta m x}g(x-\alpha n)\}_{m,n\in {\mathbb {Z}}}. \end{aligned}$$ G ( g ; α , β ) : = { e 2 π i β m x g ( x - α n ) } m , n ∈ Z . In particular, we prove that for Herglotz windows g such systems always form a frame for $$L^2({\mathbb {R}})$$ L 2 ( R ) if $$\alpha ,\beta >0$$ α , β > 0 , $$\alpha \beta \le 1$$ α β ≤ 1 . For general rational windows $$g\in L^2({\mathbb {R}})$$ g ∈ L 2 ( R ) we prove that $${\mathfrak {G}}(g;\alpha ,\beta )$$ G ( g ; α , β ) is a frame for $$L^2({\mathbb {R}})$$ L 2 ( R ) if $$0<\alpha ,\beta $$ 0 < α , β , $$\alpha \beta <1$$ α β < 1 , $$\alpha \beta \not \in {\mathbb {Q}}$$ α β ∉ Q and $${\hat{g}}(\xi )\ne 0$$ g ^ ( ξ ) ≠ 0 , $$\xi >0$$ ξ > 0 , thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of $$L^2({\mathbb {R}})$$ L 2 ( R ) .