So I'm working this HW problem, namely(adsbygoogle = window.adsbygoogle || []).push({});

Suppose f is a continuous function on [itex]\mathbb{R}[/itex], with period 1. Prove that

[tex]\lim_{N\rightarrow\infty} \frac{1}{N}\sum_{n=1}^{N} f(\alpha n) = \int_{0}^{1} f(t) dt[/tex]

for every real irrational number [itex]\alpha[/itex].

The above is for context. The hint says to "Do it first for [itex]f(t)=\exp(2\pi ikt),k\in\mathbb{Z}[/itex]," and I have done so. I supposed that the hint pointed to using the Fourier series for f. My question is: since f is continuous, may I assume that the Fourier series for f converges uniformly to f? [I recall something about the Gibbs phenomenon that made me ask.]

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# Homework Help: Hilbert space: Fourier Series

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