Zeros of the Zak transform of totally positive functions

2014/11/06 by Tobias Kloos, Kloos, Tobias · 1 citation
Engineering · Mathematics · #30E10 #41A15 #42C15 #42C40 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1411.1539

openalex publication_date 2014/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Zak transform of totally positive (TP) functions. We use the convergence of the Zak transform of TP functions of finite type to prove that the Zak transforms of all TP functions without Gaussian factor in the Fourier transform have only one zero in their fundamental domain of quasi-periodicity. Our proof is based on complex analysis, especially the Theorem of Hurwitz and some real analytic arguments, where we use the connection of TP functions of finite type and exponential B-splines.

Citations

• The wavelet transform, time-frequency localization and signal analysis

Cited by

• Gabor Frames of Totally Positive Functions: A Complete Characterization

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