Fourier uniqueness in even dimensions
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2021-04-13
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Journal Article
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Abstract
In recent work, methods from the theory of modular forms were used to obtain Fourier uniqueness results in several key dimensions ([Formula: see text]), in which a function could be uniquely reconstructed from the values of it and its Fourier transform on a discrete set, with the striking application of resolving the sphere packing problem in dimensions [Formula: see text] and [Formula: see text] In this short note, we present an alternative approach to such results, viable in even dimensions, based instead on the uniqueness theory for the Klein-Gordon equation. Since the existing method for the Klein-Gordon uniqueness theory is based on the study of iterations of Gauss-type maps, this suggests a connection between the latter and methods involving modular forms. The derivation of Fourier uniqueness from the Klein-Gordon theory supplies conditions on the given test function for Fourier interpolation, which are hoped to be optimal or close to optimal.
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published
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118 (15)
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National Academy of Sciences
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Subject
Fourier transform; Fourier uniqueness; Heisenberg uniqueness pairs; Klein-Gordon equation
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02500 - Forschungsinstitut für Mathematik / Institute for Mathematical Research
02500 - Forschungsinstitut für Mathematik / Institute for Mathematical Research