Chebyshev-Markov-Stieltjes inequalities

In mathematics, The Chebyshev–Markov–Stieltjes inequalities are important inequalities related to the problem of moments. They allow to extract some information about the measure from its first moments; namely, they provide sharp bounds on the measure of a halfline.

Formulation

Let $c\_0,\; c\_1,\; dots,\; c\_\{2m-2\}\; in\; mathbf\{R\}$; consider the collection C of measures $mu$ on $mathbf\{R\}$ such that $int\; x^k\; dmu(x)\; =\; c\_k$ for "k" = 0,1,...,2"m"-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 $mu$, and let $xi\_1,\; dots,\; xi\_m$ be the zeros of $P\_m$.

It is not hard to see that the polynomials $P\_0,\; dots,\; P\_\{m-1\}$ and the numbers $xi\_1,\; dots,\; xi\_m$ are defined uniquely by $c\_0,\; c\_1,\; dots,\; c\_\{2m-2\}$.

Define $ho\_\{m-1\}(z)\; =\; 1\; Big/\; sum\_\{k=0\}^\{m-1\}\; |P\_k(z)|^2$.

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

: $mu(-infty,\; xi\_j]\; leq\; ho\_\{m-1\}(xi\_1)\; +\; cdots\; +\; ho\_\{m-1\}(xi\_j)\; leq\; mu(-infty,xi\_\{j+1\}).$

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.

ee also

* Truncated moment problem

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.