info prev up next book cdrom email home

Separation Theorem

There exist numbers $y_1<y_2<\ldots<x_{n-1}$, $a<y_{n-1}$, $y_{n-1}<b$, such that


where $\nu=1$, 2, ..., $n$, $y_0=a$ and $y_n=b$. Furthermore, the zeros $x_1$, ..., $x_n$, arranged in increasing order, alternate with the numbers $y_1$, ...$y_{n-1}$, so


More precisely,


for $\nu=1$, ..., $n-1$.

See also Poincaré Separation Theorem, Sturmian Separation Theorem


Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 50, 1975.

© 1996-9 Eric W. Weisstein