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says
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Homework Statement
Find the quadratic least squares Chebyshev polynomial approximation of:
g(z) = 15π/8 (3-z^2)√(4-z^2) on z ∈ [-2,2]
Homework Equations
ϕ2(t) = c0/2 T0(t) +c1T1(t)+c2T2(t)
T0(t)=1
T1(t)=t
T2(t)=2t2-1
Cj = 2/π ∫ f(t) Tj(t) / (√(1-t2) dt
where the bounds for the integration are -1 to 1.
The Attempt at a Solution
The first thing I know I need to do is transform into t [-1,1], so:
t = z-1 ⇒ z = t+1
at z = 0, t = -1
at z = 2, t = +1
t ∈ [-1,1]
g(z) = (15π/8) (3-z2)√(4-z2)
g(t) = (15π/8) (3-(t+1)2)√(4-(t+1)2)
I'm a bit confused at this point. Hoping someone can be of help. How I've proceeded is below
2/π ∫ g(t) Tj(t) / (√(1-t2)) dt
bounds of integration are again from -1 to 1
c0 = 2/π ∫ (15π/8) (3-(t+1)2)√(4-(t+1)2) (1) / (√(1-t2)) dt
c1 = 2/π ∫ (15π/8) (3-(t+1)2)√(4-(t+1)2) (t) / (√(1-t2)) dt
c2 = 2/π ∫ (15π/8) (3-(t+1)2)√(4-(t+1)2) (2t2-1) / (√(1-t2)) dt
Once i evaluate c0, c1, c2 I can then put them in the equation ϕ2(t) and then sub the z value back in and get the Chebyshev polynomial approximation. At least I think that's how it's done.