Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I know that there exist continuous function $f: [0,2\pi]\rightarrow\mathbb{R}$ whose Fourier series diverges at all rational points of $[0,2\pi]$(c.f. Katznelson).We also know that the set of divergence of $f$ must be of measure zero, from Carleson, since a continuous function is undoubtedly in $L^2([0,2\pi])$. But there is a gap: can a continuous function have its Fourier series diverges on an uncountable set of measure zero(like the Cantor set)?

share|improve this question

1 Answer 1

up vote 9 down vote accepted

Kahane and Katznelson proved that given any set $E$ of measure zero, there exists a continuous function whose Fourier series diverges on $E$ (see this paper).

share|improve this answer
    
thanks you sir, i get it now –  Wilson of Gordon Apr 26 at 10:55

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.