# Set of continuous functions dense in C[-pi,pi]

• November 9th 2011, 11:59 AM
Chandru1
Set of continuous functions dense in C[-pi,pi]
How do i show that the set of continuous functions in $C[-\pi,\pi]$ which are periodic, that is $f(+\pi)=f(-\pi)$ is dense in $L^{2}[-\pi,pi]$
• November 9th 2011, 10:25 PM
CaptainBlack
Re: Set of continuous functions dense in C[-pi,pi]
Quote:

Originally Posted by Chandru1
How do i show that the set of continuous functions in $C[-\pi,\pi]$ which are periodic, that is $f(+\pi)=f(-\pi)$ is dense in $L^{2}[-\pi,\pi]$

Consider the convergence of Fourier series of functions in $L^{2}[-\pi,\pi]$?

(since the partial sums of the Fourier series are in $C[-\pi,\pi]$)

CB
• November 9th 2011, 10:51 PM
Chandru1
Re: Set of continuous functions dense in C[-pi,pi]
Quote:

Originally Posted by CaptainBlack
Consider the convergence of Fourier series of functions in $L^{2}[-\pi,\pi]$?

(since the partial sums of the Fourier series are in $C[-\pi,\pi]$)

CB

Can you give a full proof of this. I am not aware of anything. A reference page in an internet would be good.
• November 10th 2011, 04:19 AM
CaptainBlack
Re: Set of continuous functions dense in C[-pi,pi]
Quote:

Originally Posted by Chandru1
Can you give a full proof of this. I am not aware of anything. A reference page in an internet would be good.

Googleing for "l2 convergence of fourier series" will give you several PDFs which discuss this, alternativly follow the links in the Norm Convergence area of the Wikipedia page:

Convergence of Fourier series - Wikipedia, the free encyclopedia

CB