Multipoint Geronimus and Schur parameters of measures on a circle and on a line

A theorem of Geronimus that a measure corresponding to a Carathéodory function with sufficiently small Schur parameters has a support coinciding with the whole unit circle is established in the multipoint version, when the points of interpolation of the continued fraction representing the Carathéodory function have a limit distribution (in Geronimus's classical theorem all points of interpolation are concentrated at the origin). The Geronimus and Schur parameters of measures with support on the real line are introduced. For measures with support on the real line and the corresponding Nevanlinna function it is shown that an analogue of Geronimus's theorem holds, as well as analogues of some other results on measures with support on the unit circle.
Bibliography: 18 titles.