# Fourier Coefficients of Continuous functions are square summable.

1. Apr 15, 2012

### Kreizhn

1. The problem statement, all variables and given/known data

If $C^1(\mathbb T)$ denotes the space of continuously differentiable functions on the circle and $f \in C^1(\mathbb T)$ show that
$$\sum_{n\in\mathbb Z} n^2 |\hat f(n)|^2 < \infty$$
where $\hat f(n)$ is the Fourier coefficient of f.

3. The attempt at a solution

Since f is continuous it is integrable and so $\widehat{f'}(n) = in \hat f(n)$. Thus
$$\sum_n n^2 |\hat f(n)|^2 = \sum_n |\widehat{f'}(n)|^2$$
so this boils down to showing that the Fourier coefficients of a continuous function are square summable.

Now not all continuous functions need to have absolutely convergent Fourier series, so somehow this result implies that all continuous functions have square-summable series? This is not clear to me.