Proof of exactness of Gaussian-Laguerre quadrature integration

Question

The Laguerre polynomials $a_{0}(x), a_{1}(x), a_{2}(x), \dots$ form an orthogonal set on $[0, ∞)$ and satisfy:

$\int_{0}^{\infty} e^{-x} a_{i}(x) a_{j}(x) d x=0, \quad i \neq j$

The polynomial $a_{n}(x)$ has n distinct zeros $x_{1}, x_{2}, \dots, x_{n} \text { on }[0, \infty)$. Construct a n-point quadrature formula that evaluates:

$I=\int_{0}^{\infty} e^{-x} f(x) d x$

and has precision 2n − 1.

How do I prove the statement that the quadrature formula has precision 2n − 1?

Answer 1

You don't really need to write a specialized proof for this particular case, especially since they didn't give you a particular value of $n$. The general argument basically goes like this:

• the degree of exactness is at most $2n-1$: $a_n^2$ is a polynomial of degree $2n$ whose integral is obviously positive but which the rule integrates as zero
• the degree of exactness is at least $2n-1$: given a polynomial $p$ of degree at most $2n-1$, write $p=q a_n + r$ by polynomial long division. Then check that the $q a_n$ term doesn't contribute to the integration (because of how $a_n$ was chosen) nor to the quadrature rule (because of how the nodes were chosen). Thus the problem is reduced to choosing weights to exactly integrate a polynomial of degree at most $n-1$ (namely $r$) when the nodes are forced to be the zeros of $a_n$. From this point of view you can now follow the argument for deriving a quadrature rule "of interpolation type" to obtain the weights and thus to show that a rule with the desired properties exists. (This is a terrible way to obtain the weights to actually use in numerics, but it's fine for a proof.)