Approximation of Functions, Fourier Series

To construct the trigonometric polynomial of order M of the form

$$f(x)=\frac{A_0}{2}\sum_{j=1}^{M}[A_jcos(jx)+B_jsin(jx)]$$

based on the N equally spaced values $x_k=-\pi +2\pi k/N$,for $k=1,2,\ldots,N$. The construction is possible provided that $2M+1\leq N$.
The following program constructs vectors A and B that contain the coefficients $A_j$ and $B_j$, respectively, of the equation above of order M.

function [A,B]=tpcoeff(X,Y,M)
%Input    - X is a vector of equally spaced abscisssas in [-pi, pi]
%         - Y is a vector of ordinates
%         - M is the degree of the trigomometric polynomial
%Output   - A is a vector containing the coefficients of cos(jx)
%         - B is a vector containing the coefficients of sin(jx)

N=length(X)-1;
max1=fix((N-1)/2);
if M>max1
   M=max1;
end
A=zeros(1,M+1);
B=zeros(1,M+1);
Yends=(Y(1)+Y(N+1))/2;
Y(1)=Yends;
Y(N+1)=Yends;
A(1)=sum(Y);
for j=1:M
   A(j+1)=cos(j*X)*Y';
   B(j+1)=sin(j*X)*Y';
end
A=2*A/N;
B=2*B/N;
A(1)=A(1)/2;

You are given the function $Y(X)=X$ for the interval $[-\pi ,\pi ]$.

1. Use the MATLAB program given above to calculate $A_j$s and $B_j$s.
   (Hint: You should first calculate all the $Y$ values for a given $M$, say 100.)
2. The following program will evaluate the $f(x)$ of order $M$ at a particular value of $x$. $A,B$ and $M$ values are taken from the previous item.

   function z=tp(A,B,x,M)
   z=A(1);
   for j=1:M
      z=z+A(j+1)*cos(j*x)+B(j+1)*sin(j*x);
   end
   

   Study the following commands:

   >>x=-pi:.01:pi
   >>y=tp(A,B,x,M)
   >>plot(x,y,X,Y,'o')
   

3. Repeat the procedure for the $M$-values, 10, 20, 50. Compare the results