Uniform convergence for the Fourier series of $\alpha$-Hölder periodic functions and continuous bounded variation periodic functions.

In the answers to this question it is proved that if $$f:\mathbb{R}\to\mathbb{C}$$ is a $$\alpha$$-Hölder $$2\pi$$-periodic function, then the Fourier series of $$f$$ converges uniformly to $$f$$.

In the answer to this question it is proved that if $$f:\mathbb{R}\to\mathbb{C}$$ is a continuous bounded variation $$2\pi$$-periodic function, then the Fourier series of $$f$$ converges uniformly to $$f$$.

Note that since the Weierstrass function is $$\alpha$$-Hölder continuous for every $$\alpha<1$$ and differentiable nowhere (hence not of bounded variation), and since $$x\mapsto\frac{1}{\log(x)}$$ is absolutely continuous in $$[-\frac{1}{2},\frac{1}{2}]$$ but not $$\alpha$$-Hölder continuous for any $$\alpha\in(0,1)$$, the first result doesn't imply directly the second and vice versa.

Can be both theorems be viewed as particular results of a theorem for a class of functions that contains both continuous bounded variation periodic functions and $$\alpha$$-Hölder periodic functions?

• – Calvin Khor Jan 24 at 12:39
• Thanks a lot. I'll modify the question accordingly citing your question and leaving only the second part – Bob Jan 24 at 12:41
• Of course there's no such thing as a proof that the answer is no, but I tend to think the answer's no, just because if there were such a result I suspect I would have heard about it - people would prove the two results you mention as special cases of the general result, which they don't, as far as I've ever seen. – David C. Ullrich Jan 24 at 13:38