How to Minimize the Norm of f(x)=x^3-c-bx-ax^2 Under L_2[-1,1]?

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The discussion focuses on minimizing the norm of the function f(x) = x^3 - c - bx - ax^2 under the L_2 norm over the interval [-1, 1]. Participants suggest using orthonormal functions, specifically 1, x, x^2, and x^3, to derive coefficients a, b, and c. However, the use of Legendre polynomials is recommended for accuracy in this context. One participant proposes a simpler method of integrating f^2 to find that the optimal coefficients are a = 0, c = 0, and b = 3/5.

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Assume I have the next function f(x)=x^3-c-bx-ax^2 and I am asked to find the coefficients a,b,c which minimizes the norm of f under L_2[-1,1].

All I need to do here is equate f=\sum_k <f,\phi_k>\phi_k where the phis are orthonormal functions, in this case simply 1,x,x^2,x^3, I am not sure this correct cause I found the next coefficients:
<f,1>=sqrt(-2a-2c/3)
<f,x>=sqrt(2/5-2/3 b)
<f,x^2>=sqrt(-2a/3-2c/5)
<f,x^3>=sqrt(2/7-2b/5)

But when equation I find two different solutions to b, so I suspect this is the wrong to solve this problem, any hints as to how to minimize this functional.
 
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OK I think I know why I didn't get it right, I should be using Legendre polynomial cause they are defined on this interval [-1,1].

Have I got it right this time?
 


Your approach seems unduly complicated. Why not simply integrate f2 and find the values of a, b, c which gives a minimum? I tried it myself (no guarantee - I am lousy in arithmetic) and got a=c=0 and b=3/5.
 
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