# When does a continuous function's “Fourier series” converge pointwise almost everywhere to the function?

Let $$G$$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $$L^2(G)$$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible representations of $$G$$. When $$G$$ is commutative, these irreducible representations are one-dimensional, and the Peter-Weyl theorem just says that the unitary characters of $$G$$ form an orthonormal basis of $$L^2(G)$$.

For example, when $$G = \mathbb R/\mathbb Z$$, the functions $$f_n(x) = e^{2\pi i nx}$$ are an orthonormal basis of $$L^2(G)$$. Thus if $$f \in L^2(G)$$, then there are unique complex numbers $$c_n$$, with $$\sum\limits_n |c_n|^2 < \infty$$, which gives $$f$$ its Fourier expansion:

$$f = \sum\limits_{n\in \mathbb Z} c_n f_n$$ where the sum on the right hand side converges in the $$L^2$$-norm to $$f$$. It is a much deeper theorem that when $$f$$ is continuous, the right hand side also converges pointwise to $$f$$ almost everywhere.

Is the analogue of this deep theorem known for other compact groups? That is, suppose we take an orthonormal basis $$f_i : i \in I$$ of $$L^2(G)$$ via matrix coefficients of irreducible representations as in the Peter-Weyl theorem, so that any $$f \in L^2(G)$$ has a "Fourier expansion" for uniquely determined $$c_i \in \mathbb C$$,

$$f = \sum\limits_{i \in I} c_i f_i$$ so the right hand side converges to $$f$$ in the $$L^2$$-norm. Suppose that $$f$$ is continuous. Then, do we know that the right hand side converges pointwise to $$f$$ almost everywhere?

If this is not known in general, is it known for, say, $$G = \mathbb A_k/k$$ for $$k$$ a global field?

• How are you performing your summation for a general $I$? That is: your infinite sum must be interpreted as some kind of limit, but how do you wish to do it? This is a problem even if $G={\bf T}^2$, if I recall correctly – Yemon Choi Jan 2 at 4:48
• A hint at some of the potential problems with making your question more precise can be found in the remarks on the Wikipedia page for Carleson's theorem: en.wikipedia.org/wiki/Carleson%27s_theorem – Yemon Choi Jan 2 at 4:49
• I see. Maybe the statement of my question is unclear for general $G$. – D_S Jan 2 at 4:53
• D_S, I was gently trying to point out that it's not even clear fo the particular case of ${\bf T}^2$. However, perhaps for the kinds of groups you ask about in your final line, one can say more? I am not familiar enough with Fourier analysis on those kinds of groups to hazard a guess – Yemon Choi Jan 2 at 4:56
• For $k$ a global field (which is countable), and $\mathbb A$ the adele group of $k$, the quotient group $\mathbb A/k$ is compact, and its character group identifies with $k$. So in the Fourier expansion, I would want the infinite sum over $k$ to be taken in any order, since there is no predetermined order on $k$. – D_S Jan 2 at 5:02