Hands-on-Fourier Series; Approximation of Functions with MATLAB II

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

$$x_k=-\pi+ 2\pi \frac{k}{N}, for k=1,2,\ldots,N$$

The construction is possible provided that $2M+1\leq N$. The following http://siber.cankaya.edu.tr/ozdogan/NumericalComputations/mfiles/chapter4/tpcoeff.m program constructs vectors A and B that contain the coefficients $A_j$ and $B_j$, respectively, of the equation above of order M.

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.

   
   Study the following commands:

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

3. To summarize; repeat the procedure for the $M$-values, 10, 20, 50, 100. Compare the results.
   Solution:

   
   
   save with the names calc.m and drawer.m. Then;

   >> drawer