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.

1 Formulation
2 History
3 See also
4 References

[edit] Formulation

Let $c_0, c_1, \dots, c_{2m-2} \in \mathbf{R}$ ; consider the collection $\mathfrak{C}$ of measures $μ$ on $\mathbf{R}$ such that $\int x^k d\sigma(x) = c_k$ for $k = 0,1,\dots,2m-1$ (and in particular the integral is defined and finite). Let $P_0, P_1, \dots, P_{m-1}, P_{m}$ be the first $m + 1$ orthogonal polynomials with respect to $μ$ , and let be the zeros of $P m$ .

Lemma The polynomials $P_0, \dots, P_{m-1}$ and the numbers $\xi_1, \dots, \xi_m$ are defined uniquely by $c_0, c_1, \dots, c_{2m-1}$ .

Define $\rho_{m-1}(z) = 1 \Big/ \sum_{k=0}^{m-1} |P_k(z)|^2$ .

Theorem (Ch-M-S) For $j = 1, \dots, m$ and any $\mu \in \mathfrak{C}$ ,

$\mu(-\infty, \xi_j] \leq \rho_{m-1}(\xi_1) + \cdots + \rho_{m-1}(\xi_j) \leq \mu(-\infty,\xi_{j+1}).$

[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.