This question is related to my earlier question. The only difference is that I am now asking for a lower bound. (I wish I had asked this earlier.)
I have a single variable function $f(x)$ defined for $x\in[0,L]$. The actual form of $f(x)$ is not known, but I know that $f(x)$ is infinitely differentiable and its derivatives are bounded and non-zero (so that constant function is ruled out, thanks to the comment by User8128).
Now I write the unknown $f(x)$ as a cosine series: $f(x)=a_0/2+\sum_{n=1}^\infty a_n\cos (n\pi x/L)$ in which $a_n=(2/L)\int_0^Ldx~f(x)\cos(n\pi x/L)$. I want to know if one may infer reasonably tight lower bound on $a_n$ (or at least a lower bound on $|a_n|$)? Preferably that lower bound is a function of $n$. Thanks for any help.