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Help needed regarding proof of definite integral problem

  1. Mar 27, 2013 #1
    1. The problem statement, all variables and given/known data

    f(x) is a bounded function and integrable on [a,b] . a, b are real constants. We have to prove that

    i) An = ab f(x)cos(nx) dx → 0 when n→∞
    ii)Bn = ab f(x)sin(nx) dx → 0 when n→∞


    2. Relevant equations

    Parseval's formula : For uniform convergence of f(x) with its fourier series in [a,b]

    abf2(x) dx = L * ( a02/2 + Ʃ( an2+bn2 ) )


    where a0 an bn are Fourier Coefficient. L = (b-a)/2

    3. The attempt at a solution

    If f(x) is bounded integrable in [-π,π]
    Then by Parseval's Theorem
    πf2(x) dx = L * ( A02/2 + Ʃ( An2+Bn2 ) )

    Hence Ʃ( An2+Bn2 ) ) ≤ 1/L * ( πf2(x) dx )

    Hence Ʃ( An2+Bn2 ) ) converges to a finite value.

    Hence An → 0 when n→∞ and Bn → 0 when n→∞

    The above proof is assumed the interval is [-π,π] .

    But the problem is when interval is [a,b]. Fourier coefficient An is given by
    (1/L)*ab f(x)cos(nπx/L) dx

    Any idea How to proceed further?
     
  2. jcsd
  3. Mar 27, 2013 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Just apply a scale-location transformation of the form x = A*u + B, to transform the x-interval [a,b] into the u-interval [-π,π].
     
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