I'm looking for theorems that deal with the pointwise convergence of double Fourier series expansions for a special class of functions.
Let $D \subset [-\pi, +\pi]^2$ be an arbitrary set of finite perimeter with piecewise $C^1$ boundary. Let $\mathbf{1}_D : [-\pi, +\pi]^2 \to \mathbb{R}$ be the indicator function of the set $D$, and $f_D : \mathbb{R}^2 \to \mathbb{R}$ be the periodic extension of $\mathbf{1}_D$.
My questions are:
- 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 almost everywhere to $f_D$?
- Is the sequence of rectangular (or square) partial Fourier sums uniformly bounded?
For any $L^2$ function $f$, the square partial sums converge to $f$ a.e., and so do certain (but not general) rectangular sums. This is an extension of Carleson's theorem, obtained independently (in different forms) by Ch. Fefferman, P. Sjölin, and N. Tevzadze. See the survey by J. Marshall Ash and the paper by Fefferman.
I do not have an answer to question 2. It may have a better chance of being answered at MathOverflow.