Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5773726 | Journal of Approximation Theory | 2017 | 11 Pages |
Abstract
Let α>0 and let gâL1(R) be a continuous function, whose Fourier transform is gÌ(Ï)=CeâγÏ2eâ2ÏiδÏâν=1âe2ÏiδνÏ1+2ÏiδνÏâj=1meλjâ2ÏiαÏâ1λjâ2ÏiαÏ,where C>0, γ⩾0, δ,δν,λjâR, âν=1âδν2<â, mâZ+. We prove that its Zak transform Zαg(x,Ï)=âkâZg(x+αk)eâ2ÏikÎ±Ï has only one zero (xâ,12α) in the fundamental domain [0,α)Ã0,1α. In particular, the result is valid for totally positive functions. Earlier it was known for such functions without the factor eâγÏ2. We also establish simplicity of the zero with respect to each variable and give the applications to Gabor analysis. The described class of functions is closed under convolution.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
O.L. Vinogradov, A.Yu. Ulitskaya,