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I want to prove Carleson's theorem for $L^2$: If $f\in L^2(\Bbb T)$, $$\lim_{n \to \infty}S_n(f)=f \ \ \ a.e.$$ I have learned that $$\lim_{n\to \infty}||f-S_n(f)||_2=0$$ Take limit into the norm, $||f-\lim S_n(f)||_2=0$, so $\lim S_n(f) = f$ a.e. Isn't this right? But the wikipedia says that proof is not easy.

** After seeing comments, I missed that $S_n(f)$ may not converge to $\lim S_n(f)$ in $L^2$ sense. (I'm also not sure it converges pointwisely.)

Gobi
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  • The simplest version of the $L^2$ proof of which I'm aware is by Michael Lacey and Christoph Thiele, and you can find a good explanation here. – JavaMan Jun 09 '13 at 01:32
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    Convergence in the $L_2$ norm does not imply pointwise convergence a.e., if that's what you wish. See here for a counterexample (which can be modified to work in your space). – David Mitra Jun 09 '13 at 01:37

1 Answers1

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Isn't this right?

No. As pointed out by David Mitra, the $L^2$ convergence does not imply convergence a.e.

But the wikipedia says that proof is not easy.

Wikipedia is right. If you are looking to practice your knowledge of Fourier analysis, it is better to begin with something less famous, such as the list of unanswered Fourier analysis questions.

Community
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