Let $I$ be a closed interval (bounded or not) of $\mathbb{R}$ and let $w\colon I \rightarrow \mathbb{R}$ be a continuous function that is positive inside $I$ and such that for every $n\in \mathbb{N}_0$,

$$\int_I \left| t \right |^n \cdot w(t) dt < \infty.$$

$w$ is a weight function. Let:

$$H_w=\left \{f:I \rightarrow \mathbb{R} | f \mbox{ is Lebesgue measurable and } \int_I w(x) \cdot \left | f(x) \right |^2 < \infty \right \}.$$

Let the scalar product in $H_w$ be:

$$\langle f,g\rangle=\int_a^b w(x) f(x) g(x) dx,$$ and let $(a_n)_{n \in \mathbb{N}_0}$ be the succession of polynomials $a_n(t)= t^n$ and let $(P_n)_{n \in \mathbb{N}}$ be the succession you obtain from $(a_n)_{n \in \mathbb{N}_0}$ by the following method:

$P_0=a_0,\ \, \, P_n=a_n-\mbox{ projection of }a_n \mbox{ over the subspace generated by }P_0,\dots,P_{n-1}$.

Show that if $I$ is a closed and bounded interval then the succession of polynomials $(P_n)_{n \in \mathbb{N}}$ is a orthogonal basis of the space $H_w$.

Show also that if $I=[0,\infty)$ and $w(t)=e^{-t}$ then the functions $e_n=e^{-nt}$ form a total family in $H_w$ and this the Laguerre polynomials $(P_n)_{n \in \mathbb{N}}$ form an orthogonal basis of $H_w$

I've started by proving that $H_w$ with the given scalar product is complete, but I can't proceed from here.


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