Fourier series: Parseval's identity HELP

[sarahisme]

Hey all,

I am unsure how to do this problem... i find problems where i have to derive things quite difficult! :P

http://img143.imageshack.us/img143/744/picture2ao8.png

this is the Full Fourier series i think and so the Fourier coeffiecients would be given by:

http://img144.imageshack.us/img144/8200/picture5yv4.png

ok so first i need to take the inner product, so i did this:

http://img99.imageshack.us/img99/7193/picture4ol9.png

but then i am stuck... anyone got an idea of how to proceed from here?

Cheers! :)

Sarah

 

[quasar987]

How about writing the inner product in this form instead...

$$\int_{-L}^{+L}|f(x)|^2dx = \int_{-L}^{+L} f(x) \left( \frac{A_0}{2} + \sum_{n\geq 1} A_n\cos (\frac{n\pi x}{L})+B_n\sin (\frac{n\pi x}{L}) \right)dx$$

does this inspire you more? (It should)

 

[sarahisme]

How about writing the inner product in this form instead...

$$\int_{-L}^{+L}|f(x)|^2dx = \int_{-L}^{+L} f(x) \left( \frac{A_0}{2} + \sum_{n\geq 1} A_n\cos (\frac{n\pi x}{L})+B_n\sin (\frac{n\pi x}{L}) \right)dx$$

does this inspire you more? (It should)

it does actually, hold on... i'll post my answer in a sec

 

[sarahisme]

http://img136.imageshack.us/img136/8739/picture6tt5.png

however i do have one question, are we allowed to substitute the integral inside the summation? (i am have never heard of any rules telling me whether this is or is not allowed...)

 

[quasar987]

Haven't you covered series of functions (or at least sequences of functions) in an earlier class?

 

[sarahisme]

not for a couple of years i don't think...

thanks for the help by the way :)

 

[quasar987]

Strange, since Fourier series are precisely series of functions. In occurence, sine and cosine. They are usually covered in a second analysis class together with the theory of Riemann integration.

 

[sarahisme]

yeah, we did Riemann integration about a year and half ago i think, in a first analysis class I'm pretty sure.

 

[Swapnil]

So when is it OK to move the integral inside the summation?

 

[quasar987]

Thm: If the series of function $\sum_{n=1}^{\infty} f_n(x)$ converges uniformly towards S(x) on [a,b] and if $f_n$ is integrable on [a,b] $\forall n\in \mathbb{N}$, then the function S is integrable on [a,b] and

$$\sum_{n=1}^{\infty}\int_a^b f_n(x)dx=\int_a^b \sum_{n=1}^{\infty} f_n(x)dx$$

 

[Higgs_Boson]

http://img136.imageshack.us/img136/8739/picture6tt5.png

however i do have one question, are we allowed to substitute the integral inside the summation? (i am have never heard of any rules telling me whether this is or is not allowed...)

In this case it is surely ok to include the integral in the summation. Don't know the rule exactly but I remember my professor claiming that it's ok to do the above one.

I will try to find supporting evidence in due course!

 

[maxverywell]

In this case, how do we check if the Fourier series converges uniformly to the f(x)?
Actually we don't know if the function f is continuous and C^1 in [-L,L] so I think that we can't "move the integral inside the summation". If it was saying, for example, that the function f is periodic with L=pi and f is square integrable than we could do this.