(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

for vector space C[-1,1] with L^2 inner product

<f,g> = [tex]\int[/tex]f(x)g(x)dx

find the best least squares approximation for function x^(1/3) on [-1,1] by a quadratic function q(x) = c0 + c1x + c2x^2

2. Relevant equations

s+r = n

<t^s, t^r> = [tex]\int[/tex]t^ndt = { 2/(n+1) if n is even

0 if n is odd }

3. The attempt at a solution

q(x) = c0*1 + c1*x + c2*x^2

take inner product of functions of q(x)

||1|| = sqrt(2)

||x|| = sqrt(2/3)

||x^2|| = sqrt(2/5)

normalize vectors in the basis

[tex]\hat{u1}[/tex] = 1/sqrt(2)

[tex]\hat{u2}[/tex] = x/sqrt(2/3)

[tex]\hat{u3}[/tex] = x^2/sqrt(2/5)

find coefficients by taking integrals of unit vectors with function x^1/3

c1 = (1/sqrt(2))[tex]\int[/tex]x^1/3dx = [tex]\stackrel{3}{4sqrt(2)}[/tex]

c2 = (1/sqrt(2/3))[tex]\int[/tex]x^4/3dx = [tex]\stackrel{3}{7sqrt(2/3)}[/tex]

c3 = (1/sqrt(2/5))[tex]\int[/tex]x^7/3dx = [tex]\stackrel{3}{10sqrt(2/5)}[/tex]

therefore p(x) = c1[tex]\hat{u1}[/tex] + c2[tex]\hat{u2}[/tex] + c3[tex]\hat{u3}[/tex]

just wanting to confirm my answer, thanks for any and all help anyone can give and I'll write back this time, lol

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# Homework Help: Method of Least Squares question

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