Fourier series with sigma approximation

Fourier series with sigma approximation

Program FFTSIGMA plots Fourier series representations with sigma approximation. The figures show effects of
the number of series terms and use of Lanczos sigma factors
to smooth Gibbs oscillations. The Fourier series of a periodic function with period px has the approximate
form:

f(x) = sum( exp(2i*pi/px*k*x)*c(k),...
k=-n:n)

If the function has discontinuities, a better approximation
can sometimes be produced by using a smoothed function fa(x)
obtained by local averaging of f(x) as follows:

fa(x) = integral(f(x+u)*du, -s<u<+s )/(2*s)

where s is a small fraction of px. Wherever f(x) is
smooth, f and fa will agree closely, but sharp edges of f(x)
get rounded off in the averaged function fa(x). The Fourier
coefficients ca(k) for the averaged function are simply
ca(k) = c(k)*sig(k) where the sigma factors sig(k) are
sig(k) = sin(sin(2*pi*s*k/px)*/(2*pi*s*k/px))
( SEE Chapter 4 of 'Applied Analysis' by Cornelius Lanczos )