Integration and differentiation of periodic functions

Integration and differentiation of periodic functions using FFT

% Suppose that f(t) is a real 2pi-periodic function on the interval [0,2pi]
% and the 1*n vector x is the values of the function f(t) at the n
% equidistant points (n must be even)
% t_j=(j-1)*2*pi/n, j=1,2,...,n.
% The function
% y = derfft (x)
% uses fft to find the trigonometric interpolating polynomial that
% interpolates the function f(t) at the n points t_1,t_2,...,t_n. Then the
% function derfft computes the values the periodic derivative function
% f'(t) at the above points t_j.
%
% Example
% n = 2^7;
% t = [0:2*pi/n:2*pi-2*pi/n].';
% x = sin(t).*exp(-cos(t));
% xpe = cos(t).*exp(-cos(t))+sin(t).^2.*exp(-cos(t));
% xp = derfft (x);
% error = norm(xp-xpe,inf)
%
% If the function f(t) satisfies
% \int_(0)^(2pi)f(t)dt=0 and \int_(0)^(2pi)cos(nt)f(t)dt=0
% The function
% y = intfft (x)
% uses fft to find the trigonometric interpolating polynomial that
% interpolates the function f(t) at the n points t_1,t_2,...,t_n. Then the
% function intfft computes the values a periodic antiderivative function
% g(t) at the above points t_j, i.e.,
% g(t) = \int f(t)dt
% and g(2pi)=g(0).
% Note that g(t) is only one of the antiderivatives of the function f(t)
% and hence g(t) is not unique.
%
%
% Example
% n = 2^7;
% t = [0:2*pi/n:2*pi-2*pi/n].';
% x = sin(t).*exp(-cos(t));
% xie = exp(-cos(t));
% xi = intfft (x);
% error = xi-xie %must be a constant