Bound for Fourier series

Question

Let $\{a_k\}$ be an infinite sequence, with $\sum_{k=\infty}^\infty \vert a_k\vert^2<\infty$. Let $f(\omega)=\sum_{k=-\infty}^\infty a_k e^{ik\omega}$ be its Fourier series. By Plancheral's theorem, $f$ is bounded. How do I get a reasonable bound for $|f|$, i.e., if $|f|\leq g$, how do I find $g$?

Answer 1

The fact that $\sum_{k \in Z} |a_k|^2$ is finite just means that $f(\omega)$ is an $L^2$ function, and Plancherel's theorem says that any $L^2$ function arises as $f(\omega)$ for some such sequence $\{a_k\}_{k \in Z}$. If you want $f(\omega)$ to be bounded, you have to assume further conditions on $\{a_k\}_{k \in Z}$. For example, if you know that $\sum_k |a_k|$ is finite as well, then by summing the absolute values of the terms you get that $|f(\omega)| \leq \sum_k |a_k|$.