Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.)

share|improve this question

1 Answer 1

up vote 5 down vote accepted

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.

share|improve this answer
    
@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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.