Let $f:S^1\to\mathbb{C}$ be a continuous function and define $$f_n(e^{i\theta})=\sum_{k=-n}^n\frac{1}{2\pi}\int_{-\pi}^{\pi}f(e^{it})e^{ik(\theta-t)}dt.$$ Then must $f_n\to f$ uniformly on $S^1$? I believe the answer is no because $$\sum_{k=-\infty}^\infty e^{ik(\theta-t)}=\frac{1}{1-e^{i(\theta-t)}}+\frac{1}{1-e^{-i(\theta-t)}}-1$$ which has a pole at $t=\theta$. However, I can't find an explicit counterexample. Does anyone know of one?
2026-04-03 14:05:19.1775225119
Does this sum converge uniformly?
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