Assignment 4 Fourier Series; Approximation of Functions; Due to January 07, 2008

To construct the trigonometric polynomial of order M of the form

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

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 $2M+1\leq N$. You are given the http://siber.cankaya.edu.tr/NumericalComputations/mfiles/chapter4/tpcoeff.mprogram that constructs vectors A and B that contain the coefficients $A_j$ and $B_j$, respectively, of the equation above of order M.

• Modify this program so that it will find the trigonometric polynomial of period $2P=d-c$ when data points are equally spaced over the interval $[c,d]$.
• Use your modified program to find $T_5(x)$ for
  
  $$f(x)=-x^2+9 for -3\leq x < 3$$
  
  
  using 60 equally spaced data points.
• Graph $T_5(x)$ and the data points on the same coordinate system.