A modified Schur algorithm and an extended Hamburger moment problem

Olav Njåstad
1991 Transactions of the American Mathematical Society  
An algorithm for a Pick-Nevanlinna problem where the interpolation points coalesce into a finite set of points on the real line is introduced, its connection with certain multipoint Padé approximation problems is discussed, and the results are used to obtain the solutions of an extended Hamburger moment problem. (A function F(z) which is analytic for z £ HQ+ with ImF(z) > 0 is called a Nevanlinna function.) The problem was solved by Pick [27, 28] in the case that A is finite, by Nevanlinna [15,
more » ... by Nevanlinna [15, 16] in the case that A is countable, and by Krein and Rekhtman [14] in the general case. A variant of the problem for finite or countable A arises when all points za coalesce to a single point a , and given values wa at the points are replaced by the Taylor coefficients at a. A problem closely related to this is Carathéodory's coefficient problem (see [2, 3, 33]): Given a finite sequence {y0, ... , ym} or an infinite sequence {yn : n £ N}, find a function F(z) which is analytic in the open unit disc D° = {z : \z\ < 1} such that Re7(z) > 0 for z £ D° and F(z) = EZU?*2* + EZm+\ôkm)zk. or F(z) = Er=o^z¿-(For historical remarks on this problem, see [12] .) By introducing the linear fractional transformation 7 -► j=£ we can reformulate the requirement Re7(z) > 0 to read |7(z)| < 1. (A function F(z) which is analytic for z £ D is called a Carathéodory function if Re F(z) > 0 for z € D°, or a Schur function if |7(z)| < 1 for z E 7J°.) Schur [31] invented an algorithm called the Schur algorithm to deal with this problem. The technique was adapted by Nevanlinna to deal with the Pick-Nevanlinna problem.
doi:10.1090/s0002-9947-1991-1024773-1 fatcat:2of6ogwnt5fijaybyk2v7vmqai