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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A class of extremal functions for the Fourier transform


Authors: S. W. Graham and Jeffrey D. Vaaler
Journal: Trans. Amer. Math. Soc. 265 (1981), 283-302
MSC: Primary 42A38; Secondary 10H30
DOI: https://doi.org/10.1090/S0002-9947-1981-0607121-1
MathSciNet review: 607121
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Abstract: We determine a class of real valued, integrable functions $ f(x)$ and corresponding functions $ {M_f}(x)$ such that $ f(x) \leqslant {M_f}(x)$ for all $ x$, the Fourier transform $ {\hat M_f}(t)$ is zero when $ \left\vert t \right\vert \geqslant 1$, and the value of $ {\hat M_f}(0)$ is minimized. Several applications of these functions to number theory and analysis are given.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0607121-1
Article copyright: © Copyright 1981 American Mathematical Society