Does there exist any continuous function $f:[\pi,\pi]\to \mathbb{C}$ whose Fourier series $\sum \hat{f}(n)e^{int}$ is almost every where divergent?
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15$\begingroup$ No -- this is Carleson's solution of Luzin's conjecture en.wikipedia.org/wiki/Carleson%27s_theorem $\endgroup$– Yemon ChoiCommented Oct 6, 2019 at 18:35
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