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These are two exercises on Rudin's Real and complex Analysis:

Compute $$\min_{a,b,c} \int_{-1}^1 | x^3 -a -bx -cx^2|^2 dx,$$ and find $$\max \int_{-1}^1 x^3g(x)dx$$ where $g$ is subject to the restrictions $$\int_{-1}^1g(x)dx = \int_{-1}^1 xg(x)dx = \int_{-1}^1 x^2 g(x) dx =0,$$ and $$\int_{-1}^1|g(x)|^2 dx =1.$$

In fact, finding out $$\min_{a,b,c} \int_{-1}^1 | x^3 -a -bx -cx^2|^2 dx$$ is not very difficult using elementary methods. But this does no help to the second question.

And exercise followed is:

Compute $$\min_{a,b,c} \int_0^{\infty} |x^3 -a -bx -cx^2|e^{-x} dx.$$ State and solve the corresponding maximum problem, as in the previous exercise.

I was suggested to use the method of orthonormal sets of Hilbert spaces, but I had no idea what to do.

I appreciate any help, hint or detailed. Thank you very much.

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1 Answer

Hint: Let $H = L^2([-1,1])$ be the real Hilbert space of all square integrable functions with the inner product $$ \left< f, g \right> = \int_{-1}^1 f(x) g(x) dx. $$ You want to maximize $\left< g, x^3 \right>$ subject to the constraints: $$ \left<g, 1 \right> = \left< g, x \right> = \left< g, x^2 \right> = 0, \;\;\left<g, g \right> = 1. $$ The conditions say that $g$ is orthogonal to $\{1,x,x^2\}$ and of length $1$, and you want to maximize the projection $\left<g, x^3 \right>$ of $g$ in the direction of $x^3$.

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