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 ,
[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.