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(Bartle's book; The elements of real analysis, 2nd edition, Exercise 38.u)

Suppose that $f$ and $f'$ are continuous with period $2\pi$ and that $f''$ is piecewise continuous with period $2\pi$. Show that the Fourier coefficients $a_n$, $b_n$ of $f$ are such that the series $\sum_{n=1}^{\infty} n^2 (|a_n| + |b_n|)$ is convergent.

user108219
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  • Are you sure you have read the exercise correctly? – Daniel Fischer Nov 08 '16 at 21:58
  • It might be useful to note that if $f(2\pi)=f(0)$ and $f'(2\pi)=f'(0)$, then by integration by parts $$\int_0^{2\pi} f(x)e^{inx},dx=-\frac1{n^2}\int_0^{2\pi}f''(x)e^{inx},dx$$ – Mark Viola Nov 08 '16 at 22:04
  • NOT Exactly. In the book it is assumed that $f''$ is piecewise continuous. But it seems that "period $2\pi$ means $f(2\pi)=f(0)$ and $f'(2\pi)=f'(0)$. – user108219 Nov 08 '16 at 22:11
  • Well, making the constraints on $f''$ weaker changes the problem in the wrong direction. Essentially, the version you wrote says that the series of Fourier coefficients of a continuous function converges absolutely. Which is wrong. Allowing piecewise continuous functions makes that claim not more true. – Daniel Fischer Nov 08 '16 at 22:27
  • The problem is changed. Thank you for your comments. – user108219 Nov 08 '16 at 23:02

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