Hilbert space: Fourier Series

[benorin]

So I'm working this HW problem, namely

Suppose f is a continuous function on $\mathbb{R}$, with period 1. Prove that

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

for every real irrational number $\alpha$.

The above is for context. The hint says to "Do it first for $f(t)=\exp(2\pi ikt),k\in\mathbb{Z}$," 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.]

 

[quantumdude]

My PDE book says that you may conclude that the Fourier series for $f$ converges uniformly to $f$ on $[a,b]$ if:

1.) $f$, $f'$, and $f''$ are all continuous on $[a,b]$ and,
2.) f satisfies the boundary conditions.

So continuity of $f$ alone is not sufficient to establish uniform convergence.

 

[quasar987]

I agree with the main point of your post Tom. That is, that

continuity of $f$ alone is not sufficient to establish uniform convergence.

But what boundary conditions are we talking about in condition 2. ?! Is this said in the context of the Sturm-Liouville equation that has $cos(2\pi nx/(b-a))$ and $sin(2\pi nx/(b-a))$ as its eigenfunctions? Namely, if f satisfies the same boundary conditions as the ones associated with the sturm-liouville equation that has $cos(2\pi nx/(b-a))$ and $sin(2\pi nx/(b-a))$ as a basis for its solutions, and satisfies condition 1., then the Fourier serie of f converges uniformly to f on [a,b].

What I said might not make perfect sense as I didn't do a lot of Sturm-Liouville, but I find it really fascinating.

 

[benorin]

Specifically, what are necessary and sufficient conditions that the Fourier series for f:

i. actually converge to f ?

ii. be uniformly convergent ?

iii. both i and ii ?

 

[benorin]

Did I interpet the hint correctly then? (Or are Fourier series not the way to go?)

 

[benorin]

Specifically, what are necessary and sufficient conditions that the Fourier series for f:
i. actually converge to f ?
ii. be uniformly convergent ?
iii. both i and ii ?

I found it uncomfortable quoting myself, but... I did. Anyhow,

A sufficient condition for (ii) is... if f is a periodic entire function of period 2*pi, then the [usual] Fourier series for f converges uniformly on every horizontal strip containing the real axis.