Fourier interpolation on the real line

By Danylo V. Radchenko, Maryna S. Viazovska. Publications mathématiques (Bures-sur-Yvette), 2017

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In this paper we construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set {0,±1,±2,±3,…}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{0, \pm\sqrt{1}, \pm\sqrt{2}, \pm\sqrt{3},\dots\}$\end{document}. The functions in the interpolating basis are constructed in a closed form as an integral transform of weakly holomorphic modular forms for the theta subgroup of the modular group.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2194ff68d4d14763815753dade6ef1d907b0a444