1. ## Inverse Fourier Series

Under specific conditions for a periodic function we can always find a Fourier Series for the continous points.

But is it possible to do the inverse operation? That is, given a Fourier series to find a periodical function that has its expansion. (The reason why I say "a function" because such a function will not be unique, as you know the series converges to the left-right limits at the discontinuity points. Und it is certainly possible to find another different function).

Such a techinque would prove useful when solving a partial differencial equation which leads to a two point boundary value problem. They are solved in terms of a Fourier series. But we simply have a series solution not a closed form solution.

I realize that the answer is probably not. For example, when working with a power series there is no standard way. However we can manipulate the infinite series and try to create an ordinary differencial equation out of it so we can solve for the function. But is there some useful techique for inverting a Fourier series like inverting a a power series?

2. Originally Posted by ThePerfectHacker
Under specific conditions for a periodic function we can always find a Fourier Series for the continous points.

But is it possible to do the inverse operation? That is, given a Fourier series to find a periodical function that has its expansion. (The reason why I say "a function" because such a function will not be unique, as you know the series converges to the left-right limits at the discontinuity points. Und it is certainly possible to find another different function).

Such a techinque would prove useful when solving a partial differencial equation which leads to a two point boundary value problem. They are solved in terms of a Fourier series. But we simply have a series solution not a closed form solution.

I realize that the answer is probably not. For example, when working with a power series there is no standard way. However we can manipulate the infinite series and try to create an ordinary differencial equation out of it so we can solve for the function. But is there some useful techique for inverting a Fourier series like inverting a a power series?
Take any non-elementary function (well behaved) on [0,1] (lets say
Lambert's W) it has a Fourier series, but that series does not have a closed
form representation in terms of the elementary functions (if it did so would
the W function on (0,1)).

RonL

3. Originally Posted by CaptainBlank
Take any non-elementary function (well behaved) on [0,1] (lets say
Lambert's W) it has a Fourier series, but that series does not have a closed
form representation in terms of the elementary functions (if it did so would
the W function on (0,1)).

RonL
I know. But I am asking is there some useful method to approach this problem. Some manipulation of the terms to will lead somewhere. For example, with Power series on way is to take derivatives and try to set up a differencial equation which might be solved in elementary form.