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How do we know that the limit of the partial sums of the Fourier expansion exists? Can it happen that for an integrable function limit superior and limit inferior of the partial sums are unequal at some point?

Johan Aspegren
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  • There are sufficient conditions, like Dirichlet's that, if satisfied, ensure convergence. –  Feb 12 '20 at 15:48
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    On the other hand, there are L^1 functions whose Fourier series diverges everywhere. And even the Fourier series of a continuous function can diverge at some points. – Thorgott Feb 12 '20 at 15:51
  • You are referring to pointwise convergence I presume. The limit can exist in other senses as well, for example $L^2$ convergence. – copper.hat Feb 12 '20 at 16:15
  • The pointwise convergence is proved for L^2 for example. I wan`t to know how we know that the series is well defined in some sense that I cannot exactly express. The examples I have seen have Fourier series diverging to infinity. – Johan Aspegren Feb 12 '20 at 16:42
  • I meant to say that the a.e convergence is proved for L^2. – Johan Aspegren Feb 12 '20 at 16:59
  • A difficult theorem, the Carlesson-Hunt theorem, give you almost everywhere convergence for $f \in L^p$ and $p > 1$. In $p = 1$ as Thorgott said there is a counterexample due to Kolmogorov. I gave some references already in this post: https://math.stackexchange.com/questions/3227007/how-to-understand-the-convergence-of-fourier-series-in-lp/3228215#3228215 – Adrián González Pérez Feb 14 '20 at 13:12
  • OK. If the series is well defined, from $L_p$ convergence for $p > 1$ we can trivially deduct the a.e convergence, because then there is a subsequence converging a.e. But in $L_1$ we don't have norm convergence. I was wondering can those series still be well defined in some sense, even though they can be infinite. – Johan Aspegren Jun 07 '22 at 08:36

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