Separation Theorem

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

$$\lambda_\nu=\alpha(y_\nu)-\alpha(y_{\nu-1}),$$

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

$$x_\nu<y_\nu<x_{\nu+1}.$$

More precisely,

$$\alpha(x_\nu+\epsilon)-\alpha(a)<\alpha(y_\nu)-\alpha(a)=\lambda_1+\ldots+\lambda_\nu<\alpha(x_{\nu+1}-\epsilon)-\alpha(a)$$

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

See also Poincaré Separation Theorem, Sturmian Separation Theorem

References

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