Given a function $g: [0,\pi] \to \mathbb{R}$, if for example $g(0) = g(\pi) = 0$ and we write the odd and periodic extension of $g$ as a Fourier series

$$ g(x) = \sum_{m=1}^{\infty} {\hat{g}_m \sin{m x}} $$

What we can say about the series of coeficients

$$ \sum_{m=1}^{\infty} |\hat{g}_m| \ ? $$

There are minimum conditions on $g$ that can make this series convergent?

Obviously if $g \in L^2([0, \pi])$ the Bessel's formula states

$$ {\frac 2 \pi} \int_{0}^{\pi} g^2 = \sum_{m=1}^{\infty} |\hat{g}_m|^2 $$

so $ \sum_{m=1}^{\infty} |\hat{g}_m|^2 $ converges.



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