Darboux transformations of Jacobi matrices and Padé approximation

Let J be a monic Jacobi matrix associated with the Cauchy transform F of a probability measure. We construct a pair of the lower and upper triangular block matrices L and U such that J=LU and the matrix J_C=UL is a monic generalized Jacobi matrix associated with the function F_C(λ)=λF(λ)+1. It turns out that the Christoffel transformation J_C of a bounded monic Jacobi matrix J can be unbounded. This phenomenon is shown to be related to the effect of accumulating at ∞ of the poles of the Padé approximants of the function F_C although F_C is holomorphic at ∞. The case of the UL-factorization of J is considered as well.