  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.
