A class of extremal functions for the Fourier transform

Graham, S. W.; Vaaler, Jeffrey D.

doi:10.1090/S0002-9947-1981-0607121-1

A class of extremal functions for the Fourier transform
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by S. W. Graham and Jeffrey D. Vaaler

Trans. Amer. Math. Soc. 265 (1981), 283-302

DOI: https://doi.org/10.1090/S0002-9947-1981-0607121-1

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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 $\left | t \right | \geqslant 1$, 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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