clear

%%
% This function has two distinct frequencies. 
f = @(x) 3*cos(5*pi*x) - exp(2i*pi*x);

%%
% We set up to use the built-in |fft|. Note how the definition of the nodes
% has changed. 
n = 10;
N = 2*n+1;
t = 2*(0:N-1)'/N;      % nodes in $[0,2)$
y = f(t);

%%
% We perform Fourier analysis using |fft| and then examine the
% coefficients.
c = fft(y)/N;
k = [0:n -n:-1]';    % frequency ordering for MATLAB
plot(k,real(c),'.')
axis([-n n -2 2])
grid on
xlabel('k')
ylabel('c_k')
title('Interpolant coefficients')

%%
% Note that 1.5 e^{5i pi x}+1.5 e^{-5i pi x} = 3 cos(5 pi x) by
% Euler's formula, so this result is sensible.

%%
% Fourier's greatest contribution to mathematics was to point out that every
% periodic function is just a combination of frequencies---infinitely many
% of them in general, but truncated for computational use. 
f = @(x) exp( sin(pi*x) );
c = fft(f(t))/N;
semilogy(k,abs(c),'.')
xlabel('k')
ylabel('|c_k|')
title('Fourier coefficients')
grid on

%%
% The Fourier coefficients of smooth functions decay exponentially in
% magnitude as a function of the frequency. This decay rate is directly
% linked to the convergence of the interpolation error.