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

[defunc]

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!

 

[jbunniii]

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