Chebyshev–Markov–Stieltjes Inequalities

In mathematical analysis, the Chebyshev–Markov–Stieltjes inequalities are inequalities related to the problem of moments that were formulated in the 1880s by Pafnuty Chebyshev and proved independently by Andrey Markov and (somewhat later) by Thomas Jan Stieltjes. Informally, they provide sharp bounds on a measure from above and from below in terms of its first moments.

Formulation

Given ''m''₀,...,''m''_2''m''-1 ∈ R, consider the collection C of measures ''μ'' on R such that

: $\int x^k d\mu(x) = m_k$

for ''k'' = 0,1,...,2''m'' − 1 (and in particular the integral is defined and finite).

Let ''P''₀,''P''₁, ...,''P''_''m'' be the first ''m'' + 1 orthogonal polynomials with respect to ''μ'' ∈ C, and let ''ξ''₁,...''ξ''_''m'' be the zeros of ''P''_''m''. It is not hard to see that the polynomials ''P''₀,''P''₁, ...,''P''_''m''-1 and the numbers ''ξ''₁,...''ξ''_''m'' are the same for every ''μ'' ∈ C, and therefore are determined uniquely by ''m''₀,...,''m''_2''m''-1.

Denote

:$\rho_(z) = 1 \Big/ \sum_^ , P_k(z), ^2$.

Theorem For ''j'' = 1,2,...,''m'', and any ''μ'' ∈ C,

:$\mu(-\infty, \xi_j] \leq \rho_(\xi_1) + \cdots + \rho_(\xi_j) \leq \mu(-\infty,\xi_).$

References

{{DEFAULTSORT:Chebyshev-Markov-Stieltjes inequalities
Theorems in mathematical analysis
Inequalities (mathematics)