Let $F$ be a real function $2\pi$-periodic.
What is the lowest regularity that must be imposed on $F$ to admit a development in Fourier series ?
2026-03-25 17:38:29.1774460309
Regularity and Fourier series
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To define a Fourier series for an object $f$, you must be able to give a meaning to $\left<f, e^{in(\cdot)}\right>$. This is possible for any generalized function on the circle.
Now if you want the series to converge to $f$, you need to specify the type of convergence: $L^2$, $L^1$, pointwise, etc. And each of these will have different requirements.