A trigonometric polynomial of degree n is a function of the form
P(t) = a0 + a1cos t + b1 sin t + a2 cos 2t
+ b2sin2t + ............+ an cos n t + bn sin n t.
where a0,a1.b1........an,bn are the coefficients. Find the trigonometric polynomial of degree n that is the least squares approximation to the function f(t) = 1/(1+ t2) on the interval [ - π, π] based on the k equally spaced data points

(omitting the right hand endpoint), when
(a) n = 1 ,k = 4
(b)n = 2, k = 8
(c) n = 2, k = 16
(d) n = 3,k = 16
(e) Compare the graphs of the trigonometric approximant and the function, and discuss. (0 Why do we not include the right hand endpoint tk = π?

Transcribed Image Text:

tj =-r+-, 2 j=0. . . . ,k-1,