Chebyshev–Markov–Stieltjes inequalities

In

Let P₀,P₁, ...,P_m be the first m + 1 orthogonal polynomials with respect to μ ∈ C, and let ξ₁,...ξ_m be the zeros of P_m. It is not hard to see that the polynomials P₀,P₁, ...,P_m-1 and the numbers ξ₁,...ξ_m are the same for every μ ∈ C, and therefore are determined uniquely by m₀,...,m_2m-1.

Denote

${\displaystyle \rho _{m-1}(z)=1{\Big /}\sum _{k=0}^{m-1}|P_{k}(z)|^{2}}$.

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

${\displaystyle \mu (-\infty ,\xi _{j}]\leq \rho _{m-1}(\xi _{1})+\cdots +\rho _{m-1}(\xi _{j})\leq \mu (-\infty ,\xi _{j+1}).}$

References