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Convergence of Unique Fourier Series

  1. Jun 19, 2012 #1
    Hi, I'm new to this forum, so I apologize if my LaTeX looks messed up.

    1. Find the Fourier Series for [tex]f(x) = \sqrt{|x|}[/tex] and prove it converges to [tex]f(x)[/tex]


    3. So, I've thus far proved that [tex]\sqrt{|x|}[/tex] is piecewise continuous by proving that the limit as x approaches 0 (from both the right and left) for [tex]f(x) = \sqrt{|x|}[/tex] is equal to 0. However, the function fails to be piecewise smooth because its derivative has a vertical tangent and, thus, most of the convergence theorems that I've learned (such as the Fourier series converging to the average of the right and left limits) are inaccessible.

    So, I wrote out the Fourier series myself for the arbitrary interval -a < x < a, getting:

    For Fourier coefficients, we get:

    [tex]a_0 = \frac{1}{2a}\int_{-a}^a \sqrt{|x|} dx = \frac{2}{3}\sqrt{|x|}[/tex]

    [tex]a_n = \frac{2}{a}\int_0^a \sqrt{|x|} \cos{(\frac{n\pi x}{a})}[/tex]

    [tex]b_n = 0 [/tex](because the function is even)

    The Fourier series becomes

    [tex]\sqrt{|x|} \approx \frac{2}{3}\sqrt{|x|} + \frac{2}{a}\sum_{n=1}^\infty \int_0^a \sqrt{|z|}\cos{(\frac{n\pi z}{a})} dz \cos{(\frac{n\pi x}{a})}[/tex]

    I tried to use a theorem stating that if the series [tex]\sum_{n=1}^\infty |a_n| + |b_n| [/tex] converges, then the Fourier series converges uniformly. So, essentially, I must prove
    [tex]\sum_{n=1}^\infty |\frac{2}{a}\int_0^a \sqrt{|x|} \cos{(\frac{n\pi x}{a})}|[/tex] converges.

    I tried to use an array of traditional convergence tests, but none of them seemed to work.
     
    Last edited: Jun 20, 2012
  2. jcsd
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