Convergence of Fourier Series Coefficients for L2 Functions

[Kindayr]

Homework Statement

Let e_{n}(t)= \frac{1}{ \sqrt{2\pi}}\cdot e^{int} for n\in\mathbb{Z} and -\pi\le t\le\pi.

Show that for any f\in L^{2}[-\pi,\pi] we have that (f,e_{n})=\int_{-\pi}^{\pi}f(t)\cdot e^{-int}dt\to0 as |n|\to \infty.

The Attempt at a Solution

I want to use dominant convergence, but unfortunately measure theory isn't a prerequisite for this course. Any help will be awesome!

 

[Kindayr]

Wait, if {e_n} is complete, then \lim_{n\to\infty}\sum_{m=-n}^{n}(f,e_m)e_m converges to f absolutely, so the coefficients necessarily converge to zero. Would this work?

 

[micromass]

Have you seen and proved Bessel's inequality??

 

[Kindayr]

Yep we have. Could I say that:

Since \{e_n\} is complete it follows that f=\lim_{n\to\infty} \sum_{m=-n}^{n}(f,e_m)e_m. Thus, the latter sum converges and hence \lim_{n\to\infty}\sum_{m=-n}^{n}|(f,e_m)|^{2}<\infty. Thus |(f,e_m)|^2 \to 0 so |(f,e_m)|\to 0 and whence (f,e_m)\to 0, as required.

Does this work?

 

[micromass]

Yep we have. Could I say that:

Since \{e_n\} is complete it follows that f=\lim_{n\to\infty} \sum_{m=-n}^{n}(f,e_m)e_m. Thus, the latter sum converges and hence \lim_{n\to\infty}\sum_{m=-n}^{n}|(f,e_m)|^{2}<\infty. Thus |(f,e_m)|^2 \to 0 so |(f,e_m)|\to 0 and whence (f,e_m)\to 0, as required.

Does this work?

You don't really need \{e_n\} to be complete for that. Bessel's inequality works fine.
Also, the limit you mention is L^2-convergence. You might want to be careful with that.

 

[Kindayr]

You don't really need \{e_n\} to be complete for that. Bessel's inequality works fine.
Also, the limit you mention is L^2-convergence. You might want to be careful with that.

What should I be careful with? Does L^2-convergence not imply absolute convergence? Sorry, just a little confused by your statement.

 

[Kindayr]

What if I re-index so that n\in \mathbb{N} so that d_0=e_0, d_1=e_1, d_2=e_{-1},\dots. Then by Bessel's inequality we have \sum_{n=0}^{\infty}|(f,d_n)|^2\le \parallel f\parallel ^{2}<\infty. Hence \sum_{n=0}^{\infty}|(f,d_n)|^2 converges absolutely and whence (f,d_n)\to 0 and thus (f,e_n)\to\ 0 as |n|\to\infty.

Would this be better?

I'm far from an analyst as you can probably tell hahahah.