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Separation Theorem


The separation theorem states that there exist numbers y_1<y_2<...<y_(n-1), a<y_(n-1), y_(n-1)<b, such that

 lambda_nu=alpha(y_nu)-alpha(y_(nu-1)),
(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).
(2)

More precisely,

 alpha(x_nu+epsilon)-alpha(a)<alpha(y_nu)-alpha(a)=lambda_1+...+lambda_nu<alpha(x_(nu+1)-epsilon)-alpha(a)
(3)

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


See also

Poincaré Separation Theorem, Sturmian Separation Theorem

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References

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

Referenced on Wolfram|Alpha

Separation Theorem

Cite this as:

Weisstein, Eric W. "Separation Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SeparationTheorem.html

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