Rayleigh theorem, projection of orbital measures and spline functions

Abstract We consider a random matrix X uniformly distributed on an orbit for the action of the orthogonal group on the space of real symmetric matrices or of the unitary group on the space of Hermitian matrices. The problem is to evaluate the distribution of the eigenvalues of a compression of X. We give a survey about this question and present some new results. Baryshnikov's formula and Olshanski's determinantal formula are revisited, and a Markov–Krein type formula is established.