Chebyshev-Markov-Stieltjes inequalities
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The Chebyshev--Markov--Stieltjes inequalities are important inequalities related to the problem of moments. The inequalities give sharp bounds on the measure of a halfline in terms of the first moments of the measure.
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[edit] Formulation
Let 
; consider the collection 
of measures μ on 
such that 
for 
(and in particular the integral is defined and finite). Let 
be the first m + 1 orthogonal polynomials with respect to μ, and let be the zeros of Pm.
Lemma The polynomials 
and the numbers 
are defined uniquely by 
.
Define 
.
Theorem (Ch-M-S) For 
and any 
,
![\mu(-\infty, \xi_j] \leq \rho_{m-1}(\xi_1) + \cdots + \rho_{m-1}(\xi_j) \leq \mu(-\infty,\xi_{j+1}).](../../../math/4/c/d/4cd271ab1533ae1f0cdd459e23ba9bb8.png)
[edit] History
The inequalities were formulated in the 1880-s by Pafnuty Chebyshev and proved independently by Andrey Markov and (somewhat later) by Thomas Jan Stieltjes.
[edit] See also
- Truncated moment problem
[edit] References
- Akhiezer, N. I., The classical moment problem and some related questions in analysis, translated from the Russian by N. Kemmer, Hafner Publishing Co., New York 1965 x+253 pp.
