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

I'm looking for references/theorems that deal with the pointwise convergence of double Fourier series expansions for a particular function.

Let $D \subseteq [-\pi, +\pi]^2$ be an arbitrary set of finite perimeter with piecewise $C^1$ boundary. Let $\mathbf{1}_D : [-\pi, +\pi]^2 \to \{ 0, 1 \}$ be the indicator function of the set $D$, and $f_D : \mathbb{R}^2 \to \{ 0, 1 \}$ be the periodic extension of $\mathbf{1}_D$.

My questions are:

  1. Considering rectangular (or square) partial Fourier sums, does there exist a theorem that tells us that the double Fourier series expansion of $f_D$ converges pointwise everywhere and that the pointwise limit function is almost everywhere equal to $f_D$?
  2. Is the sequence of rectangular (or square) partial Fourier sums uniformly bounded?
share|improve this question

1 Answer 1

It is known that the square partial sums of any $L^2$ (or, for that matter, $L^p$ for $1 < p \leq \infty$) function on $\mathbb{T}^2$ converges almost everywhere. This can be deduced from (the one dimensional) Carleson's theorem. This gives an affirmative answer to the (square version) of your first question. See*: C. Fefferman, On the convergence of multiple Fourier series. Bull. Amer. Math. Soc. 77 (1971), 744–745

It is known that $2$-d rectangular summation can fail to converge pointwise almost everywhere even for $L^\infty$ functions. However, it is unclear to me if the known counterexample can be modified to give a function in the form your considering above. See*: C. Fefferman, On the divergence of multiple Fourier series. Bull. Amer. Math. Soc. 77 1971 191–195.

*Confusingly, there is an error with mathscinet and the reviews for these two papers have been transposed. Update: It appears the mathscinet has now corrected the issue.

share|improve this answer
    
Thanks for the references! I think I have found out more precise results in the book ''Serie trigonometriche'' of L. Tonelli. –  Hans May 3 '13 at 14:59

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.