Fourier series with sigma approximation

Fourier series with sigma approximation

Description

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 )

Acknowledgements

The author wishes to acknowledge the following in the creation of this submission:
Double Fourier Series Surface Plotting

• MATLAB release: MATLAB 7.6 (R2008a)