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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.)

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up vote 5 down vote accepted


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

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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@Jonas. Perhaps a reference should be useful. – TCL Nov 18 '10 at 21:18
@TCL: I don't have one. I will look for it in some books on Fourier series. – Jonas Teuwen Nov 18 '10 at 22:57
@TCL: I found one, Zygmund's first book has a paragraph on Fourier series that converge pointwise but not uniformly, it is essentially the same example. – Jonas Teuwen Nov 18 '10 at 23:01
@Jonas. I search thru the whole book and couldn't find the paragraph you said. Do you mind let me know which section in which chapter? Thanks. – TCL Nov 20 '10 at 4:14
@TCL: Trigonometric series, part I, page 300 (Chapter Divergence of Fourier Series). – Jonas Teuwen Nov 20 '10 at 23:16

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