On the convergence of Padé approximations for generalized Nevanlinna functions

Abstract

We study a stepwise algorithm for solving the indefinite truncated moment problem and obtain the factorization of the matrix describing the solution of this problem into elementary factors. We consider the generalized Jacobi matrix corresponding to Magnus' continuous P -fraction that appears in this algorithm and the polynomials of the first and second kind that are solutions of the corresponding difference equation. Weyl functions and the resolution matrices for finite and infinite Jacobi matrices are computed in terms of these polynomials. Convergence of diagonal and paradiagonal Padé approximation for functions from the generalized Nevanlinna class is studied.

Boundary interpolation and rigidity for generalized Nevanlinna functions

“…It can be shown that any Θ ∈ U z1 can be written as a minimal product of elementary factors; see [2] and [20] for the case z 1 = ∞.…”

Asymptotics for polynomials orthogonal in an indefinite metric

“…Unlike the classical case, we see that for a finite number of values z ∈ D, the limit in (3.9) is zero. Indeed, if z 0 ∈ C \ R is an eigenvalue of H (and hence a singularity of the m-function; see [5,6] [16,Theorem 13.3.1] here). Then, if z ∈ D is such that z + 1/z = z 0 , then the limit in (3.9) is zero.…”

Jacobi matrices generated by ratios of hypergeometric functions

“…Then, if we consider the bilinear form on ℓ 2 (x, y) G = (Gx, y) ℓ2 , x, y ∈ ℓ 2 , we see that (Hx, y) G = (x, Hy) G . Next, following [9] and [10] we can also introduce the m-function of H via the formula m(z) = ((H − z) −1 e, e) G , e = (1, 0, 0, . .…”