Fourier Series-Several questions

[WannaBe22]

Homework Statement

1. Prove that if f(x) is continuous in R with a period of 2pi and hjer Fourier coefficients are 0 then
f(x)=0. Deduce that two different continuous functions in R with a period of 2pi has different Fourier series..

2. Prove by finding the Fourier series at (0,pi) that for every x in (0,pi):
cosx= 8/pi * Sigma [ (n*sin(2nx) ) / (4n^2 - 1) ]. Check if the formula is correct for x=0 and x=pi and explain why the series doesn't uniformly converges at (0,pi). Is it pointwise converge at (0,pi)? Mean converges?

Homework Equations

The Attempt at a Solution

I've no idea how to solve it...I'm pretty lame at this and have no idea what to do... I'll be glad to receive some detailed guidance...

Thanks in advance

 

[HallsofIvy]

"Fourier factors"? Do you mean Fourier coefficients? For (1) you should know an "error" formula for the error when approximating a Fourier series by a finite partial sum.

For (2) just use the usual integral formula to expand cos(x) in a Fourier sine series. As far as "uniform convergence" is concerned, it should be clear from the fact that cos(0)= 1 while sin(n(0))= 0 for all n. As for pointwise convergence, again look at x= 0.

 

[WannaBe22]

Yep, I meant Fourier coefficients ...I didn't quite understand the way to solve (1)...

About 2- Very understandable...Tnx

I'll be glad to get some further guidance...

Thanks a lot

 

[WannaBe22]

Well thanks a lot...I've re-read your answer nd it's completely understandable now...