I'm trying to prove that if $|\widehat{f}(k)| = |a_k| < \frac{M}{|k|^{2 + \varepsilon}}$ then $f$ is continuously differentiable. I'm not quite sure how to do this. There was another question on the site (Rate of decay of Fourier coefficients vs smoothness) but it makes an incorrect assumption: that we can termwise differentiate the Fourier series. Here is what I've done so far.
It's obvious that $\sum a_k e^{ikx}$ converges uniformly to some function $g$ and that $g \equiv f$ almost everywhere, with $g$ continuous. So we need to show that $g$ is continuously differentiable. The continuity part is easy once we have differentiability, so we just need to show differentiability. To that end we have
$\lim_{y \to x} \frac{g(x)-g(y)}{x-y} = \lim_{y \to x} \frac{\sum a_k e^{ikx} – \sum a_k e^{iky}}{x-y} = \lim_{y \to x} \sum a_k \left( \frac{e^{ikx} – e^{iky}}{x-y} \right) = \sum a_k \lim_{y \to x} \left( \frac{e^{ikx} – e^{iky}}{x-y} \right) = \sum a_k ike^{ikx}$
which converges based on our bound on $a_k$. However, I don't really know how to prove the third equality. I don't think that the inner functions are monotone in absolute value, and I can't think of any dominating function. Can someone point me in the right direction?
Best Answer
Hint: Use the theorem
Let $g_n$ in this case be the Fourier partial sums $\sum_{|k| \leq n} a_k e^{ikx}$.