I have this orthonormal system of polynomials: $$L_{n}(x):=\frac{e^{x}}{n!}\frac{d^{n}}{dx^{n}}(e^{-x}x^{n})$$ And I suppose that this is a basis in $L^{2}(0,+\infty)$ endowed whit the s.p. $(u,v):=\int_{0}^{\infty}u(x)v(x)e^{-x}dx$.
I have to compute the polynomial of degree $\leq 2$ which approximate the function $$f(x)=e^{x/4}$$ In $L^{2}(0,+\infty)$ endowed with the above defined s.p. $(u,v)$.
Any suggestion to start? Thank you very much.