Fourier Series-Several questions

1. Jan 9, 2010

WannaBe22

1. The problem statement, all variables and given/known data

1. Prove that if f(x) is continous in R with a period of 2pi and hjer fourier coefficients are 0 then
f(x)=0. Deduce that two different continous 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?

2. Relevant equations
3. 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 recieve some detailed guidance...

Last edited: Jan 9, 2010
2. Jan 9, 2010

HallsofIvy

Staff Emeritus
"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.

3. Jan 9, 2010

WannaBe22

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