Example 4.  Given the 12  equally spaced data points

which can be extended periodically over , if we define  .  
Find the Fourier polynomial of degree n = 5  for the 12  equally spaced points over interval  .  
Use Mathematica's Fit subroutine to find the coefficients.

Solution 4.

Note.  The data are computed using the function  , which is the function that was used in examples 1 - 3.  

Construct the data points to be used.

Remark.  Notice that precisely 12 data points are used and a point corresponding to is not used.   

Remark.  The Mathematica procedure  Fit  can be used to numerically compute the trigonometric polynomial.

Caveat.  The underlying function f(x) is even which requires that only cosine terms be used for the Fourier trigonometric polynomial.
Since f(x) has period , if data points were used at both end of the interval then the "least squares" algorithm would weight the function value twice, whereas all other function values would have weight 1.  If 13 data points were used which included the right end point, then a wrong answer will result, and spurious terms appear in the trigonometric polynomial. Look at the wrong answer.

Wrong Answer.

(c) John H. Mathews 2004