hmmm, thanks for the info. i've mostly worked with simple examples, where f(x) is peicewise continuous, and where each piece is linear. you can just plot the graph of the sum of the first 100 terms of the fourier series, then read the equations of the pieces off the graph. one example did have a piece that was a polynomial, and curve-fitting told me that it was a quadratic. it's all pretty ad-hoc, though. bob --- Bill Gosper wrote:
Bob Baillie asked
One of the first examples of Fourier series that a student encounters is something like this: f(x) = x/2, for -Pi < x < Pi.
The Fourier series is Sum[ (-1)^(n+1)/n * Sin[n x] ].
A second, more complicated example is: f(x) = -Pi/2, for -Pi < x < 0, f(x) = Pi/2, for 0 < x < Pi.
This Fourier series is Sum[ (1 - (-1)^n)/n * Sin[n x] ].
Can one work backwards, from the coefficients to the function?
About all you can do is replace sin(x) with Im(e^ix) and treat the problem as a hypergeometric series, of which you can only do the easiest cases. Rarely one might use a bibasic series with q=e^ix and p=1. To see the hopelessness of the general case, note that the Snowflake and more general fractals have fairly nice Fourier coefficients: http://gosper.org/fst.dvi, with (poorly scanned) figures http://gosper.org/fst1.png, http://gosper.org/fst1.png, http://gosper.org/fst1.png, and http://gosper.org/fst1.png . I have even seen a fairly simple Fourier series for the boundary of the Mandelbrot set. Just imagine writing a closed form for Im(Snowflake).
That paper also gives the Fourier series for an arc repeated around the edges of a regular polygon or "star", and by letting the dimension parameter d=1, the Fourier series for the polygon itself. By studying these, we can learn to recognize the series for piecewise linear functions. --rwg PS, on the Lisp Machine I once had a demo where Macsyma computed in closed form the F.s. of a point alternately tracing an equilateral triangle and its inscribed circle, which I then animated as a sum of rotating vectors, using that machine's superior color graphics primitives.
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