Does the Fourier coefficients of a function $f\in H^1(0,L)$ (the first order Sobolev space) are absolutely summable?

Question

My precise question: Let $f\in H^1(0,L)$ and let $\{f_n\}$ be its Fourier sine series coefficients on $(0,L)$, is it true that $\{f_n\}\in l^1$, i.e. $$\sum_{n}|f_n|< \infty .$$ Thanks

Answer 1

The fact that $f$ is in $H^1$ implies that its derivative $Df$ is in $L^2$. The Fourier coefficients of $Df$ are (up to a constant) $\{n |f_n|\}$. The Parseval relationship then implies that $ \{n |f_n|\} \in \ell_2$.

Since $ \{n |f_n|\} \in \ell_2$ the Cauchy-Schwarz inequality implies

\begin{align*} \sum_n |f_n| &= \sum_n n^{-1} n |f_n| \\ &\leq \left( \sum_n n^{-2} \right)^{1/2}\left( \sum_n n^2 |f_n|^2 \right)^{1/2}\\ &<\infty \end{align*}