Approx. Theory &its Appl., 9:4, Dec. 1993 9 9 9 OVERCONVERGENCE AND ZEROS OF SEQUENCES OF RATIONAL FUNCTIONS M. Simkani* ( University of Michigan-Flint, U.S.A.) Received June 5, 1992 Abstract In the present paper we consider sequences of rational functions with a bounded number or free poles con- verging uniformly with a maximum geometric o~le to a holomorphic function on a regular set, and we examine the limiting dL~tribution of the Zeros and its relations with the phenomenon of overconvergence. Our results further extend the well known classical theory of overconvergence and the zeros of sections of Taylor series. 1. Introduction The phenomenon of overconvergence and its close relations with the distribution of the zeros of sections of Taylor series were first noted by porteri~qin 1906. The theory was further developed by Jentzsch [v'6l, Ostrowski [9"1~ SzegO [141, Carlson, P61ya, Bourion 121, and others. Later Mosesson and Walsh [161 studied the analogues for the case of maximally convergent sequences of polynomials, and for the case of sequences of analytic functions respectively. In the present paper we examine sequences of rational functions, and in par- ticular we extend the results of Edrei I~l on the zeros of Pade polynomials, the results of Blatt, Saff, and the present author Iq, and the results of Prokhorov l~21 on the zeros of se- quences of rational functions of best approximation, the results of GrothmanniSlon the ze- ros of maximally convergent sequences of polynomials, and the reults of Kovacheva Isl on the zeros of sequences of rational functions of near best approximation. Our results may al- so be applied to the sequences of rational functions of interpolation. Let E ~ C be a regular set, in the sense that E is closed and bounded, and its com- plement G: = ~E ~E is connected and has the classical Green/s function g(z,oo) with pole at infinity. The function g(z,~) is harmonic in G~{~}, continuous in G-~{oo}, and zero on aG, and the function logJzJ-g(z,oo) is harmonic at infinity. The Greenls * The research was conducted while visiting Jagiellonian University, Cracow, Poland; and was sup- ported (in part) by a grant from the Faculty Development Fund of the University of M ichigan-Flint.