:confused:Show that the trigonometric series

∑1/log(n)*sin(nx) n from 2 to infinity

converges for every x, yet it is not the fourier series of a riemann integrable

function.

thanks

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- September 20th 2007, 10:36 PMXingyuana problem about fourier series
:confused:Show that the trigonometric series

∑1/log(n)*sin(nx) n from 2 to infinity

converges for every x, yet it is not the fourier series of a riemann integrable

function.

thanks - September 21st 2007, 01:23 AMOpalg
Use Abel's test to show that the complex power series converges on the unit circle (and therefore so does its imaginary part ).

I'm not sure how to do the second part of the question. If this series were the Fourier series of a*square*-integrable function then you could use the Parseval theorem to say that converges (which it doesn't). But if you are only allowed to assume that the function is integrable then this idea won't work.