Estimating Maximum of Truncated Fourier Series

Question

Let $L$ be an integer. Is there any good way to estimate the $L^{\infty}$ norm of $$f(x)=a_0+\sum_{n=1}^{2L}\Big(a_n\cos\Big(\frac{n\pi x}{L}\Big)+b_n\sin\Big(\frac{n\pi x}{L}\Big)\Big)?$$ The trivial way would be to use the triangle inequality and the fact that $|\cos(x)|, |\sin(x)|\le 1$ to get $$\|f\|_{\infty}\le |a_0|+\sum_{n=1}^{2L}(|a_n|+|b_n|).$$ One can do slightly better with Holder to get $$\begin{aligned} \|f\|_{\infty} &\le |a_0|+\sup_{x\in\mathbb{R}}\left(\sum_{n=1}^{2L}a_n^2\sum_{n=1}^{2L}\cos^2\Big(\frac{n\pi x}{L}\Big)\right)^{1/2}+\sup_{x\in\mathbb{R}}\left(\sum_{n=1}^{2L}b_n^2\sum_{n=1}^{2L}\sin^2\Big(\frac{n\pi x}{L}\Big)\right)^{1/2} \\ &\le |a_0|+\sqrt{2L}\left(\sum_{n=1}^{2L}a_n^2\right)^{1/2}+\sqrt{2L}\left(\sum_{n=1}^{2L}b_n^2\right)^{1/2}. \end{aligned}$$ Is there a way to do better?

By differentiating we also know that at whichever $x_0$ the maximum occurs, we'll have $$\sum_{n=1}^{2L}a_n\sin\Big(\frac{n\pi x_0}{L}\Big)=\sum_{n=1}^{2L}b_n\cos\Big(\frac{n\pi x_0}{L}\Big).$$

Answer 1

No, in general there's no "reasonable" way to get a "good" estimate on $||f||_\infty$ from the Fourier coefficients. If there were a lot of things would be much simpler.

That trivial estimate is very simple, but it's typically a very bad estimate. You should note that your "better" estimate is actually never any better and is usually much worse, since $$\sum_{j=1}^{2L}|c_j|\le\sqrt{2L}\left(\sum_{j=1}^{2L}|c_j|^2\right)^{1/2}.$$