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Eigenfunctions of the Fourier transform with specified zeros

Part of: Discontinuous groups and automorphic forms Geometry of numbers Harmonic analysis in several variables

Published online by Cambridge University Press:  15 February 2021

AHRAM S. FEIGENBAUM ,
PETER J. GRABNER  and
DOUGLAS P. HARDIN
AHRAM S. FEIGENBAUM
Affiliation:
Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN37240, U.S.A. e-mail: ahram.s.feigenbaum@Vanderbilt.Edu
PETER J. GRABNER
Affiliation:
Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24, 8010Graz, Austria. e-mail: peter.grabner@tugraz.at
DOUGLAS P. HARDIN
Affiliation:
Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN37240, U.S.A. e-mail: doug.hardin@Vanderbilt.Edu
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Abstract

Eigenfunctions of the Fourier transform with prescribed zeros played a major role in the proof that the E8 and the Leech lattice give the best sphere packings in respective dimensions 8 and 24 by Cohn, Kumar, Miller, Radchenko and Viazovska. The functions used for a linear programming argument were constructed as Laplace transforms of certain modular and quasimodular forms. Similar constructions were used by Cohn and Gonçalves to find a function satisfying an optimal uncertainty principle in dimension 12. This paper gives a unified view on these constructions and develops the machinery to find the underlying forms in all dimensions divisible by 4. Furthermore, the positivity of the Fourier coefficients of the quasimodular forms occurring in this context is discussed.

MSC classification

Primary: 11F03: Modular and automorphic functions 11H31: Lattice packing and covering 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Secondary: 11F11: Holomorphic modular forms of integral weight
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 2 , September 2021 , pp. 329 - 367
Copyright
© Cambridge Philosophical Society 2021

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Footnotes

†

The research of these authors was supported, in part, by the U. S. National Science Foundation under grant DMS-1516400.

‡

This author is supported by the Austrian Science Fund FWF project F5503 part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”.

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