Chebyshev polynomial approximation

[says]

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

C_j = 2/π ∫ f(t) T_j(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) T_j(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.

 

[Orodruin]

t = z-1 ⇒ z = t+1

at z = 0, t = -1
at z = 2, t = +1
t ∈ [-1,1]

Your problem statement states that z ∈ [-2,2], not [0,2].

Also, what you have done is to project the function on the Chebyshev polynomials, which is not really the method of least squares (although, of course, the result is the same). The method of least squares would be to consider the norm squared of $f-T$, where $T$ is some linear combination of the Chebyshev polynomials and then minimise this with respect to the coefficients of the linear combination.

 

[says]

so at z=-2, t=-3
z=2, t=1

that's if:
t=z-1, z=t+1
t ∈ [-3,1]

I don't see how this part is relevant to the problem as i never use it. That is unless t is supposed to be ∈ [-1,1] and I'm supposed to find a different euation for z that satisfies t ∈ [-1,1]?

 

[Orodruin]

The Chebyshev polynomials are relevant for the interval [-1,1] so you must find a different mapping from z to t.