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Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a subsequence $\{N_{i}\}$ such that $ F_{N_{i}}f \rightarrow f\ \text{ a.e.}$ as $i \rightarrow \infty$. On the other hand, Carleson's difficult and important theorem asserts, $ F_{N}f \rightarrow f\ \text{ a.e.}$ as $N \rightarrow \infty.$

My question is, why is it so important to have the result about the whole sequence where the result about the subsequence is easy to prove? It will be helpful if anyone can point out applications where it is necessary to know about the almost-everywhere convergence of the full series, i.e the knowledge about the subsequence does not suffice.

Davide Giraudo
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Swarnendu Sil
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    This doesn't answer your question, but note that a lot of mathematical theorems are proved 'just to know', not because they turn out to be 'important later'. So Carleson probably just really wanted to know if this result was true or not, and banged his head against the wall until he found a proof. – Frank Mar 25 '14 at 22:17
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    @Frank It wasn't just Carlson who wanted to know. I think it was considered a big open problem at the time. – Stephen Montgomery-Smith Mar 26 '14 at 00:50
  • @StephenMontgomery-Smith Firstly, I was being tongue-in-cheek. Secondly, this comment makes my point 'stronger'. – Frank Mar 26 '14 at 00:51
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    @Frank yes, I meant to make your comment "stronger." – Stephen Montgomery-Smith Mar 26 '14 at 00:53
  • $L^2(\mathbb R)?$ – zhw. Feb 12 '18 at 19:00

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I figured out why it is important to have the convergence of the full sequence instead of the existence of a convergent subsequence, so thought I answer my own question since nobody answered it yet. Actually, this is due to the fact that since we are talking about a series, the whole sequence of partial sums must converge for the series to be convergent. Convergence of a subsequence of the partial sums implies nothing about the series, as can easily be checked in the elementary example of the series $\sum\limits_{n} (-1)^{n}.$ There are at least two subsequences of partial sums, the odd partial sums and the even partial sums which converge ( in fact, constant), but the series clearly does not.

Swarnendu Sil
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