The separation theorem states that there exist numbers
,
,
, such that
![lambda_nu=alpha(y_nu)-alpha(y_(nu-1)),](/images/equations/SeparationTheorem/NumberedEquation1.svg) |
(1)
|
where
,
2, ...,
,
and
.
Furthermore, the zeros
, ...,
, arranged in increasing order, alternate with the numbers
,
...
,
so
![x_nu<y_nu<x_(nu+1).](/images/equations/SeparationTheorem/NumberedEquation2.svg) |
(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)](/images/equations/SeparationTheorem/NumberedEquation3.svg) |
(3)
|
for
,
...,
.
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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