I have the Inner product: $ <f, f> = \sqrt{\int_{-\infty}^{\infty} f(x)^2 \cdot e^{-x^2}}$
and I want to estimate $H_3 = 8x^3 - 12x$ with a function $g(x) = ax^2 + bx + c$ such that the norm $|| H_3(x) - g(x) ||$is minimal.
I tackled it with Least squares method - I Chose the Hermite Polynomials as a base because they are orthogonal with this given inner product, so I get: $$ \begin{pmatrix}<H_0,H_0>&<H_0,H_1>&<H_0,H_2>\\ <H1,H_0>&<H_1,H1>&<H1,H_2>\\ <H2,H_0>&<H2,H1>&<H2,H2>\end{pmatrix} \begin{pmatrix}a\\ b\\ c\end{pmatrix} =\begin{pmatrix}<f,H_0>\\ <f,H_1>\\ <f,H_2>\end{pmatrix} $$
and since $ f(x) = H_3(x) $ I get that the right side of the equation is the zero vector ($\int_{-\infty}^{\infty} H_n H_m e^{-x^2}dx ~ =~ C_n~\delta_{nm}$ where $C_n$ is just some number). and since:
$a, b, c = 0, 0,0$
But that doesn't make sense (to me) that the best estimation of $H_3$ is the zero Polynomial.
Any hints would be appreciated!