An indeterminate rational moment problem and Carathéodory functions

Let { αn }n = 1∞ be a sequence of points in the open unit disk in the complex plane and letB0 = 1 and Bn (z) = underover(∏, k = 0, n) frac(over(αk, -), | αk |) frac(αk - z, 1 - over(αk, -) z), n = 1, 2, ...,(over(αk, -) / | αk | = - 1 when αk = 0). We put L = span { Bn : n = 0, 1, 2, ... } and we consider the following "moment" problem:. Given a positive-definite Hermitian inner product 〈 ·, · 〉 in L, find all positive Borel measures ν on [- π, π) such that〈 f, g 〉 = ∫- ππ f (ei θ) over(g (ei θ), -) d ν (θ) for f, g ∈ L .We assume that this moment problem is indeterminate. Under some additional condition on the αn we will describe a one-to-one correspondence between the collection of all solutions to this moment problem and the collection of all Carathéodory functions augmented by the constant ∞. © 2007 Elsevier B.V. All rights reserved. ispartof: Journal of Computational and Applied Mathematics vol:219 issue:2 pages:359-369 ispartof: location:FRANCE, Lille status: published