Can Fourier series be differentiated term by term in the spectral method?

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In the spectral method, Fourier series can be differentiated term by term, but this does not hold true universally. The convergence of the differentiated series to the real derivative is not guaranteed in general cases. A counterexample is provided in a linked PDF, highlighting specific conditions where termwise differentiation is permissible. Understanding these conditions is crucial for proper application of Fourier series in analysis. Overall, caution is advised when differentiating Fourier series term by term.
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Hi

In the spectral method, the Fourier series is differentiated term by term. How do we know this series converges uniformly to the real derivative? Or can it be shown in general for fouries series?

Thanks a lot!
 
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It is NOT true in general that you can differentiate a Fourier series term by term. There is a counterexample at this link (PDF file) along with some conditions under which termwise differentiation is OK:

http://www.mth.pdx.edu/~daescu/mth410_510w/Fourier_series.pdf
 
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