A review of the Separation Theorem of Chebyshev-Markov-Stieltjes for polynomial and some rational Krylov subspaces

The accumulated quadrature weights of Gaussian quadrature formulae constitute bounds on the integral over the intervals between the quadrature nodes. Classical results in this concern date back to works of Chebyshev, Markov and Stieltjes and are referred to as Separation Theorem of Chebyshev-Markov-Stieltjes (CMS Theorem). Similar separation theorems hold true for some classes of rational Gaussian quadrature. The Krylov subspace for a given matrix and initial vector is closely related to orthogonal polynomials associated with the spectral distribution of the initial vector in the eigenbasis of the given matrix, and Gaussian quadrature for the Riemann-Stielthes integral associated with this spectral distribution. Similar relations hold true for rational Krylov subspaces. In the present work, separation theorems are reviewed in the context of Krylov subspaces including rational Krylov subspaces with a single complex pole of higher multiplicity and some extended Krylov subspaces. For rational Gaussian quadrature related to some classes of rational Krylov subspaces with a single pole, the underlying separation theorems are newly introduced here.