Continuous Dependence of Szego Kernel on a Weight of Integration

The weighted Szego kernel was investigated in a few papers (see Nehari in J d'Analyse Mathematique 2:126-149, 1952; Alenitsin in Zapiski Nauchnykh Seminarov LOMI 24:16-28, 1972; Uehara and Saitoh in Mathematica Japonica 29:887-891, 1984; Uehara in Mathematica Japonica 42:459-469, 1995). In all of these, however, only continuous weights were considered. The aim of this paper is to show that the Szego kernel depends in a continuous way on a weight of integration in the case when the weights are not necessarily continuous. A topology on the set of admissible weights will be constructed and Pasternak's theorem (see Pasternak-Winiarski in Studia Mathematica 128:1, 1998) on the dependence of the orthogonal projector on a deformation of an inner product will be used in the proof of the main theorem.