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I've read many statements in my textbook that include something along the lines of "the Fourier series of $f$ converges absolutely, and hence uniformly to $f$". I'm not sure why this is true.

If the Fourier series converges absolutely, then it converges. That is, $ \lim_{n \to \infty} S_n(\theta) $ converges for each $\theta$. Why does it follow then that this limit is uniform?

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Let $$S_n(\theta)=\sum_{|k|\leq n}c_ke^{2i\pi k\theta}$$ be the partial Fourier series.

Now, suppose that for all $\theta \in [0, 2\pi]$, we have absolute convergence $S_n(\theta)\rightarrow f(\theta)$.

Let $\varepsilon>0$. By definition of the series being absolutely convergent, $$\sum_{k\in\mathbb Z}|c_k|<+\infty$$ Therefore, the partial sums are Cauchy: Let $\varepsilon>0$, there exists $N>0$ such that if $n>N$ and $m>N$, $$\sum_{|k|>n,m}|c_k|<\varepsilon$$ So for all $\theta \in [0, 2\pi]$, $$\left|S_n(\theta)-S_m(\theta)\right| \leq \sum_{|k|>n,m}|c_k| < \varepsilon$$ So $\{S_n\}_{n\in\mathbb N}$ is uniformly Cauchy, so it converges uniformly.

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  • $\begingroup$ Can you clarify how we go from the fact that the series is absolutely convergent to being able to enforce a bound of $\epsilon$? $\endgroup$ – slothropp Mar 5 at 10:02

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