1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Fourier Coefficients of Continuous functions are square summable.

  1. Apr 15, 2012 #1
    1. The problem statement, all variables and given/known data

    If [itex] C^1(\mathbb T) [/itex] denotes the space of continuously differentiable functions on the circle and [itex] f \in C^1(\mathbb T) [/itex] show that
    [tex] \sum_{n\in\mathbb Z} n^2 |\hat f(n)|^2 < \infty[/tex]
    where [itex] \hat f(n) [/itex] is the Fourier coefficient of f.

    3. The attempt at a solution

    Since f is continuous it is integrable and so [itex] \widehat{f'}(n) = in \hat f(n) [/itex]. Thus
    [tex] \sum_n n^2 |\hat f(n)|^2 = \sum_n |\widehat{f'}(n)|^2 [/tex]
    so this boils down to showing that the Fourier coefficients of a continuous function are square summable.

    Now not all continuous functions need to have absolutely convergent Fourier series, so somehow this result implies that all continuous functions have square-summable series? This is not clear to me.
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted