Properties of Zeros

Since $a_0,a_1,\ldots ,a_{n-1}$ are the solutions of real linear equations, they must be real. $\Pi$ must have at least one real zero in (a,b) at which it changes sign, since otherwise $\Pi (x)w(x)$ would have a constant sign in (a,b) so that $\int _a^b\Pi (x)w(x)dx\not= 0$.$\Pi$ has n zeros altogether; suppose it changes sign at $t\leq n-1$of these, corresponding to the polynomial factor $\alpha (x)=\sum _{k=0}^t\alpha _kx^k$, and write $\Pi (x)=\alpha (x)\beta (x)$, where the polynomial factor $\beta (x)$contains the remaining n-t zeros (where there is no sign change, or the zero is outside (a,b)). For $s=0,1,\ldots t$

$$\begin{array} {lcl}0 & = & \int _a^bx^s\Pi (x)w(x)dx\\ & = & \int _a^bx^s\alpha (x)\beta (x)w(x)dx. \end{array}$$

Multiplying the sth of these by $\alpha _s$ for $s=0,1,\ldots ,t$ and adding, we find

$$\begin{array} {lcl}0 & = & \int _a^b\sum _{s=0}^t \alpha _sx... ...int _a^b\left[ \alpha (x)\right] ^2\beta (x)w(x)dx.\end{array}$$

Since the integrand is non-negative, we deduce that $\alpha$ is the zero polynomial, which in turn means that $\Pi$ is the zero polynomial. Thus our supposition that $t\leq n-1$ is contradicted and so $t\geq n$.Since $\pi$ is of degree n, we must have t=n, so $\beta$ is a constant and $\alpha$ contains all the zeros, which are thus real, distinct and in (a,b).

Theorem 3988

The Gaussian quadrature coefficients all have the same sign as the weight function w.