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.