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.)
Pointwise but not uniform convergence of a Fourier series
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analysis
fourier-analysis
fourier-series
1 Answers
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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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0@Jonas. Perhaps a reference should be useful. – 2010-11-18
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0@TCL: I don't have one. I will look for it in some books on Fourier series. – 2010-11-18
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0@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. – 2010-11-18
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0@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. – 2010-11-20
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0@TCL: Trigonometric series, part I, page 300 (Chapter Divergence of Fourier Series). – 2010-11-20