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Suppose that $f$ is continuous and has (absolute) summable Fourier coefficients: $$ \sum_{k \in \mathbb{Z}}\vert c_k \vert < \infty. $$ Is it then true that the Fourier series for $f$ converges to $f$ pointwise?

I have tried to search on this site and on Google for an answer but couldn't find it. I believe that at least it does converge but I am not sure that it has to be to $f$.

I am not necessarily looking for a proof, just a clarification on whether or not this is true.

Jacobiman
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    It seems likely that the poster's odd phrase "(absolute) summable Fourier coefficients" is meant in the sense that the Fourier series $\frac {a_0} {2} + \sum\limits_{k=1}^{n} (a_k \cos kt + b_k \sin kt)$ has the property that $\sum\limits_{k=1}^{n} (|a_k| +| b_k|) < \infty$. If so the series converges uniformly to the function. – B. S. Thomson Apr 24 '22 at 22:03
  • @B.S.Thomson: this is an unrelated question, but since I realize you participate here (MSE) and I got introduced to tour books (with Buckner^2( I would like to ask if there is (or if you remember) an easy solution to this problem in your book I found a not so trivial solution resorting to the machinery of ergodic theory, but it still bugs me that there might be a much simpler solution. Sorry for intruding like the. – Mittens Apr 24 '22 at 23:33
  • @OliverDíaz Don't remember what we had in mind. The hint plus Egorov's theorem is most likely what we intended. (That is no gurantee that it was correct.) – B. S. Thomson Apr 24 '22 at 23:40
  • @B.S.Thomson: Thanks fo your answer, that is what I thought (that the problem is more complicated than initially thought). – Mittens Apr 25 '22 at 00:18
  • @B.S.Thomson I have clarified what I meant with absolute summable Fourier coefficients – Jacobiman Apr 25 '22 at 14:07
  • @StefanLafon: now that the OP has clarified his posting, you may want to consider rewriting your answer. maybe a few edits. – Mittens Apr 25 '22 at 15:03

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