I was put on hold 2 times already for this question, I don't know how to solve it (If i knew how to solve it I wouldn't be bothering you ) and I don't know why it doesn't fit the rules of this site or why I'm being put on hold. If you could at least tell me that I would be very happy :)
So, I have the following question :Given the recurrence relation $$\pi_{k+1}(t)=(t-\alpha_k)\pi_k-\beta_k\pi_{k-1}(t)$$, k=0,1,2... for the ortogonalpolynomials (monica) {$\pi_j(^.;d \lambda)$} and defining $$\beta_0=\int_R d\lambda(t)$$ show that $$||\pi_k||^2=\beta_0\beta_1....\beta_k , k=0,1,2...$$ How can this be exploited in a practical implementation of the approximation in the least - squares sense relatively to the orthogonal system?
Which I have tried to solve
Because $\pi_{k+1}(t)-t\pi_{k}(t)$ is a polynom with the degree $\le$k it can be expressed as a linear combination of $\pi_{1},\pi_{2},...,\pi_{k}$
$\pi_{k+1}(t)-t\pi_{k}(t)=-\alpha_k\pi_k(t) - \beta_k \pi_{k-1}(t)+\sum_{j=0}^{k-2}\gamma_j\pi_j(t) $ we multiply it with $\pi_k$ and obtain
$$(-t\pi_k,\pi_k)=-\alpha_k(\pi_k,\pi_k)$$
$\alpha_k $= $\frac{(t\pi_k,\pi_k)}{(\pi_k,\pi_k)} $ k=0,1,2,...
$\pi_{k+1}(t)-t\pi_{k}(t)=-\alpha_k\pi_k(t) - \beta_k \pi_{k-1}(t)+\sum_{j=0}^{k-2}\gamma_j\pi_j(t) $ we multiply it with $\pi_{k-1}$ and obtain $$(-t\pi_k,\pi_{k-1})=-\beta_k(\pi_{k-1},\pi_{k-1})$$ Because $(-t\pi_k,\pi_{k-1})=(\pi_{k},\pi_{k-1})$ and $t\pi_{k-1}$ differs from $\pi_{k}$ by a polynom with the degree $\le k$ we obtain by orthogonality $(t\pi_k,\pi_{k-1})=-\beta_k(\pi_{k},\pi_{k})$, so $$\beta_k = \frac{(\pi_k,\pi_k)}{(\pi_{k-1},\pi_{k-1})}, k=0,1,2,...$$
This is how far I've got. I still don't know how to show that $$||\pi_k||^2=\beta_0\beta_1....\beta_k , k=0,1,2...$$
It is rather straightforward. Define, $\pi_0$ first. Let $\pi_0=\sqrt{\beta_0}$. (Generally, measures are taken as unit measures so that $\pi_0=1$ which seems better). Then, using $\beta_k = \frac{(\pi_k,\pi_k)}{(\pi_{k-1},\pi_{k-1})}$ for $k=1$ you find $\beta_1$. Since you have $||\pi_1||^2=\beta_0\beta_1$ at this step, using $\beta_k = \frac{(\pi_k,\pi_k)}{(\pi_{k-1},\pi_{k-1})}$ for $k=2$ you have $||\pi_2||^2=\beta_0\beta_1\beta_2$. For a general $k$, just use induction.
For a gentle and at the same time comprehensive introduction for orthogonal polynomials on $\mathbb{R}$ (which also answers what you asked), I can suggest the Chapter 1 of the book "Szegö's Theorem and Its Descendants: Spectral Theory for $L^2$ Perturbations of Orthogonal Polynomials" by Barry Simon.