Let $f$ be the $2\pi$ periodic function which is the even extension of $$x^{1/n}, 0 \le x \le \pi,$$ where $n \ge 2$.
I am looking for a general theorem that implies that the Fourier series of $f$ converges to $f$, pointwise, uniformly or absolutely.
Best Answer
I found the following theorems from the book "Introduction to classical real analysis" by Karl R. Stromberg, 1981.
(Zygmund) If $f$ satisfies a Hölder (also called Lipschitz) condition of order $\alpha\gt 0$ and $f$ is of bounded variation on $[0,2\pi]$, then the Fourier series of $f$ converges absolutely (and hence uniformly). p. 521.
This applies to the example in my question.
If $f$ is absolutely continuous, then the Fourier series of $f$ converges uniformly but not necessarily absolutely. p. 519 Exercise 6(d) and p.520 Exercise 7c.
(Bernstein) If $f$ satisfies a Holder condition of order $\alpha\gt 1/2$ , then the Fourier series of $f$ converges absolutely (and hence uniformly). p.520 Exercise 8 (f)
(Hille) For each $0<\alpha\le 1/2$, there exists a function that satisfies a Holder condition of order $\alpha$ whose Fourier series converges uniformly, but not absolutely. p.520 Exercise 8 (f)