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Fourier Series-Several questions

  1. Jan 9, 2010 #1
    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...

    Thanks in advance
     
    Last edited: Jan 9, 2010
  2. jcsd
  3. Jan 9, 2010 #2

    HallsofIvy

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    Staff Emeritus
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    "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.
     
  4. Jan 9, 2010 #3
    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
     
  5. Jan 11, 2010 #4
    Well thanks a lot...I've re-read your answer nd it's completely understandable now...
     
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