Solvability condition for a boundary value interpolation problem of Loewner type

Abstract

In its classical form, a boundary value interpolation problem of Loewner type requires finding a complex function ~p, analytic and having a nonnegative imaginary part in the open upper half-plane, with prescribed radial boundary values and estimates of radial derivatives at points of a given finite subset of the real axis. (We call such problems boundary value interpolation problems of Pick-Nevanlinna type). R. Nevanlirma [16] considered problems of this form, with the unit disc instead of the upper half-plane, and gave an iterative solution. It turns out that nonnegativity of a matrix constructed by interpolating data (the so-called Pick matrix) is a necessary and sufficient condition for such a problem to have a solution, analogously to interior interpolation problems of Pick-Nevanlinna type [ 15], [ 18]. K. Loewner [14] considered the problem in which radial boundary values and radial derivatives of the unknown function are prescribed at points of an interval of the real axis. (Such problems, with prescribed radial derivatives instead of estimates on a finite or infinite subset of the real axis, will be called boundary value interpolation problems of Loewner type.) Somewhat different boundary problems were studied by M. Rosenblum and J. Rovnyak [ 19]. They prescribed the radial limits of the desired function almost everywhere on a Borel subset of the real axis. Since I. P. Fedchina [8] introduced directional (or tangential) matricial Pick-Nevanlinna (interior) interpolation problems, there have appeared a number of papers on directional matricial boundary value interpolation problems: M. Rosenblum and J. Rovnyak [20], J. A. Ball [3], J. A. Ball and J. W. Helton [4], I. V. Kovalishina [13], J. A. Ball, I. C. Gohberg and L. Rodman [5], V. Bolotnikov [6], etc. Here, the unknown function is a matrix function and what is prescribed are not the full values (radial limits) but rather the values in directions of certain