Suppose $f(x)$ and $\frac{df}{dx}$ are piecewise smooth. Prove that the Fourier cosine series of the continuous function $f(x)$ can be differentiated term by term. Can anyone help me with this question?
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Does the continuous function mentioned in the second sentence have anything to do with $f$ or $f'$ from the first sentence? – user127096 Mar 08 '14 at 22:24
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I have edited my post – Roos Jansen Mar 09 '14 at 08:28
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Begin by expanding $f'$ into a Fourier sine series. (Note: if $f$ is an even function, then $f'$ is odd.) Consult your textbook / lecture notes/ online resources for the following: the Fourier series of a piecewise smooth function (such as $f'$) converges uniformly. This allows us to integrate the series term by term. This results in cosine series for $f$, concluding the proof.
In a nutshell: it is easier to interchange integration and summation than differentiation and summation. This is why the proof works from $f'$ to $f$, not from $f$ to $f'$.