I need to know an example of a function defined on the interval [0, pi] which cannot be expanded as a Fourier series. Thanks.

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- May 1st 2008, 04:17 AMcurvatureA question on Fourier series
I need to know an example of a function defined on the interval [0, pi] which cannot be expanded as a Fourier series. Thanks.

- May 1st 2008, 06:09 AMCaptainBlack
- May 1st 2008, 06:41 AMcurvature
Thanks. But I was thinking of an example of function with infinitely many discontinuities or with infinitely many extrema, if any.

- May 1st 2008, 09:29 AMThePerfectHacker
- May 1st 2008, 09:36 AMCaptainBlack
A Fourier series converges to the function a.e. so the zero Fourier series converges to this a.e. as $\displaystyle \mathbb{Q}$ is a null set. That is this function does have a Fourier series and it converges to the function a.e. as required.

(even the Fourier series of a function with a step discontinuity does not converge to the function at the step without a bit of jiggery pokery about what the value of the function is at the step)

RonL - May 1st 2008, 11:54 PMcurvature
Thanks. I know the function is called the Dirichlet function and I guess this bounded function cannot be expressed into a Fourier series, but not quite sure since no books I read tell the fact.

I wonder if examples like this motivated Dirichlet to propose the Dirichlet Conditon for Fourier series?