JOURNAL OF APPROXIMATION THEORY 55, 205-219 (1988) Equiconvergence of Some Sequences of Complex Interpolating Rational Functions* (Quantitative Estimates and Sharpness) M. A. BOKHARI Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, M&an, Pakistan Communicated by V. Totik Received August 14, 1986 1. INTRODUCTION In his classic book on interpolation and approximation, Walsh [ 1] has shown that approximation in the sense of least squares by polynomials is intimately connected with Taylor series and he suggested that “approximation in the sense of least squares by more general rational functions may also be connected with interpolation in points related to the poles of the rational functions.” A number of his theorems [ 1, Chapter IX] justify this assertion. Recently, Saff and Sharma [2] took the cue and proved a theorem which further supplements the above statement of Walsh. More precisely, let A,, p > 1, be the class of functions analytic in JzI< p but not in Izl< p. For a given integer m > -1 and for c > 1 and f~ A,, let R n+ ,,Jz, f) be a rational function of the form R n+m,nkf) :=~,+,,,(z,f)l(z"-~"), (~,+,,,(z,f)~Tr+,), (1.1) which interpolates f in the (n + m + 1)th roots of unity. For v = 0, 1,2, . . . let (cf. [2, (3.7)]) f(z)= g {~}"rn+,&&), "SO ".rn (1.2) * This research is a part of the author’s Ph.D. Thesis [S] submitted at the University of Alberta. 205 0021-9045/88 S3.00 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector