Hilbert space: Fourier Series

benorin

So I'm working this HW problem, namely

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.]

Tom Mattson

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

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

So continuity of [itex]f[/itex] alone is not sufficient to establish uniform convergence.

quasar987

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

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

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.