8.901 Suppose that w(x) is a nonnegative real function of a real variable x. Let (a, b) be a fixed interval on the x-axis. Let us suppose further that, for n = 0, 1, 2, …, the integral

${\int}_{a}^{b}{x}^{n}w(x)\text{d}x$

exists and that the integral

${\int}_{a}^{b}w(x)\text{d}x$

is positive. In this case, there exists a sequence of polynomials p_{0}(x), p_{1}(x), …, p_{n}(x), …, that is uniquely determined by the following conditions:

1. p_{n}(x) is a polynomial of degree n and the coefficient of x^{n} in this polynomial is positive.

2. The polynomials ...

Start Free Trial

No credit card required