Bessel's inequality with condition on the derivative

Question

Suppose $F: R \to C$ and $f \in C^1$.

If $\int_0^1 |f^{'}|^2 dx\le 1$

1. show $\sum_{n\neq 0}|\hat f(n)|^2\le\frac{1}{4\pi^2}$

• I know that from bessel's inequality $\sum_{n\neq 0}|\hat f(n)|^2 \le ||f||^2$ but How do I use the information about $\int_0^1 |f^{'}|^2$ ?

2. show that there is $c\in C$ so that $\int_0^1 |f(x)-c|^2dx < \frac{1}{4\pi^2} $

• I guess its related to 1. but really don't know which way to go on this one.

any help would be great, thanks!

Answer 1

$\hat f' (n)= 2.\pi. i n \hat f(n)$. So ${\vert \hat f(n)\vert }^2 \leq {1\over 4 \pi^2} {\vert \hat f'(n)\vert }^2$ (with equality iff $n=1$, and the result follows by Bessel.

Choose $c$ so that $\hat f(0)=c$, and apply Parseval to the function $f-c$, we get the result.