trigintpoly

The function trigintpoly uses fft to find the trigonometric interpolating polynomial

% 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 , yp , ypp] = trigintpoly (x,s)
% 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 trigintpoly computes the values of the functions f(t), f'(t),
% and f''(t) at the points s (s is an m*1 vector of points), i.e.
% y = f(s), yp=f'(s), ypp=f''(s)
%
%
% Example 1:
% n = 100;
% t = 0:2*pi/n:2*pi-2*pi/n;
% x = cos(2.*t).^3;
% s = [-pi/4,0,pi/2];
% [y , yp , ypp] = trigintpoly (x,s);
%
% The output of this example will be:
% y =
% 4.0326e-17 1 -1
% yp =
% 2.7187e-15 4.5406e-15 2.3184e-15
% ypp =
% -1.9042e-13 -12 12
%
%
% Example 2:
% x = [2,sqrt(2),0,-sqrt(2),-2,-sqrt(2),0,sqrt(2)];
% s = [-pi,0,pi/2,2*pi];
% [y , yp , ypp] = trigintpoly (x,s);
%
% The output of this example will be:
% y =
% -2 2 1.2246e-16 2
% yp =
% 2.4493e-16 0 -2 4.8986e-16
% ypp =
% 2 -2 -1.2246e-16 -2
%