TY - JOUR

T1 - Zeros of the Zak Transform of averaged totally positive functions

AU - Vinogradov, O. L.

AU - Ulitskaya, A. Yu

PY - 2017/10/1

Y1 - 2017/10/1

N2 - Let α>0 and let g∈L1(R) be a continuous function, whose Fourier transform is ĝ(ω)=Ce−γω2 e−2πiδω∏ν=1∞ [Formula presented] where C>0 γ⩾0 δ,δν,λj∈R ∑ν=1 ∞δν 2<∞ m∈Z+. We prove that its Zak transform Zαg(x,ω)=∑k∈Zg(x+αk)e−2πikαω has only one zero (x∗, Formula presented] in the fundamental domain [0,α)×0, [Formula presented]. In particular, the result is valid for totally positive functions. Earlier it was known for such functions without the factor e−γω2 . 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.

AB - Let α>0 and let g∈L1(R) be a continuous function, whose Fourier transform is ĝ(ω)=Ce−γω2 e−2πiδω∏ν=1∞ [Formula presented] where C>0 γ⩾0 δ,δν,λj∈R ∑ν=1 ∞δν 2<∞ m∈Z+. We prove that its Zak transform Zαg(x,ω)=∑k∈Zg(x+αk)e−2πikαω has only one zero (x∗, Formula presented] in the fundamental domain [0,α)×0, [Formula presented]. In particular, the result is valid for totally positive functions. Earlier it was known for such functions without the factor e−γω2 . 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.

KW - Exponential B-splines

KW - Gabor frames

KW - Totally positive functions

KW - Zak transform

UR - http://www.scopus.com/inward/record.url?scp=85026197366&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2017.06.001

DO - 10.1016/j.jat.2017.06.001

M3 - Article

AN - SCOPUS:85026197366

VL - 222

SP - 55

EP - 63

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

SN - 0021-9045

ER -