Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a 2$\pi$ periodic function such that $\exists$ $C>0$ and $\epsilon>0$ with $|f(x)-f(y)|\leq C|x-y|^{.5+\epsilon}$. Show that the the Fourier series of $f$ converges uniformly. There is the added hint that for general $\epsilon>0$, $\int_0^{2\pi}|f(x+h)-f(x)|^2dx$ should be calculated two different ways.
The hint leads me to recognize $\int_0^{2\pi}|f(x+h)-f(x)|^2dx<2C\pi h^{1+\epsilon}$ which seems useless by itself. This is a qual question that's had me stumped for awhile. I am assuming it involves looking for uniform boundedness over the family of partial sums of the Fourier series, but I'm feeling a little uncomfortable, and looking for assistance.
