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Separation Theorem

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

\begin{displaymath}
\lambda_\nu=\alpha(y_\nu)-\alpha(y_{\nu-1}),
\end{displaymath}

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

\begin{displaymath}
x_\nu<y_\nu<x_{\nu+1}.
\end{displaymath}

More precisely,


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

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.




© 1996-9 Eric W. Weisstein
1999-05-26