[Math] Pointwise but not uniform convergence of a Fourier series

analysisfourier analysisfourier series

What is an example of a continuous, or even better, differentiable, $2\pi$ (or 1) periodic function whose Fourier series converges pointwise but not uniformly? (Such function cannot be of Hölder class, or absolutely continuous.)

Best Answer

Consider:

$$f_{n,N}(z) = \sin(Nx) \sum_{k = 1}^n \frac{\sin(kx)}{k}$$

Now consider

$$\sum_k \frac{1}{k^2} f_{2^{k^3}, 2^{k^3 - 1}}(z)$$

Now for $x = \pi / (4n)$ and $N = 2n$ we get that

$$\sin(\pi/4) \sum_1^n \frac{1}{k} > \frac{1}{\sqrt{2}} \log n$$

So we have for some $x$ that

$$|s_{n_k + 1} - s_{n_k}| \geq \frac{1}{\sqrt{2}} \frac{1}{k^2} \log n_k$$

So we cannot have uniform convergence. I believe this is due to Hugo Steinhaus.

I hope I didn't make a mistake, but it is along these lines, I can correct it if I made an error.

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