# Fourier Series - proving function is continuous

1. Jun 16, 2010

### ramdayal9

1. The problem statement, all variables and given/known data
Let f be an integrable, periodic function whose Fourier coefficients satisfy $\sum_{-\infty}^{\infty} n^6 |\hat{f}(n)|^2 < \infty$. Prove that f is continuous.

2. Relevant equations

Looking at my notes, the only relevent things i have for this question (i think) are Bessel's inequality (but that requires f to be continuous....), the Riemann-Lebesgue Theorem, Parseval's equality and also the decay and regularity of fourier series. The statement that the uniform limit of a sequence of continuous functions is continuous, maybe I can use this?

3. The attempt at a solution

I thought this question would be simple, but I'm finding that its not. I have tried 2 methods:
1) $\sum_{-\infty}^{\infty} n^6 |\hat{f}(n)|^2 < \infty$ implies $n^6 |\hat{f}(n)|^2 \rightarrow 0$ hence $|\hat{f}(n)| \rightarrow 0$. I need to show that f is continuous i.e. the converse of the Riemann Lebesgue Lemma. How would I proceed from here?
2) using L^2 convergence theorem (parseval's), $\sum_{-\infty}^{\infty} |c_n|^2 \< \infty$ hence there exists function g such that $\hat{g}(n)=c_n=n^3 |\hat{f}(n)|$. Is this the route I should proceed?

The hypothesis also states that f is integrable, but i dont know how to use this here! Im also thinking of the M test, but dont know if I can apply it to a sum like this... Any help will be appreciated!