I was wondering if there is any characterization of compactly supported functions by their Fourier series, that is, a counterpart of the Paley-Wiener Theorem for Fourier series. More precisely, if $f\in L^2([-\pi,\pi])$ is a function defined on the closed interval $[-\pi,\pi]$ with Fourier series $\{a_n\}^\infty_{n=-\infty}$, are there any sufficient and necessary conditions for $\{a_n\}^\infty_{n=-\infty}$ to make $f$ supported inside $(-\pi,\pi)$?
2026-05-15 07:37:10.1778830630
Characterization of Fourier Series of Compactly Supported Functions
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