Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Polynomial approximation with varying weights on compact sets of the complex plane
HTML articles powered by AMS MathViewer

by Igor E. Pritsker PDF
Proc. Amer. Math. Soc. 126 (1998), 3283-3292 Request permission


For a compact set $E$ with connected complement, let $A(E)$ be the uniform algebra of functions continuous on $E$ and analytic interior to $E.$ We describe $A(E,W),$ the set of uniform limits on $E$ of sequences of the weighted polynomials $\{W^n(z)P_n(z)\}_{n=0}^{\infty },$ as $n \to \infty ,$ where $W \in A(E)$ is a nonvanishing weight on $E.$ If $E$ has empty interior, then $A(E,W)$ is completely characterized by a zero set $Z_W \subset E.$ However, if $E$ is a closure of Jordan domain, the description of $A(E,W)$ also involves an inner function. In both cases, we exhibit the role of the support of a certain extremal measure, which is the solution of a weighted logarithmic energy problem, played in the descriptions of $A(E,W).$
  • H. Alexander, Polynomial approximation and analytic structure, Duke Math. J. 38 (1971), 123–135. MR 283244, DOI 10.1215/S0012-7094-71-03816-6
  • H. Alexander, Polynomial approximation and hulls in sets of finite linear measure in Cn, Amer. J. Math. 93 (1971), 65–74. MR 284617, DOI 10.2307/2373448
  • Robert George Blumenthal, Holomorphically closed algebras of analytic functions, Math. Scand. 34 (1974), 84–90. MR 380423, DOI 10.7146/math.scand.a-11508
  • P. B. Borwein and Weiyu Chen, Incomplete rational approximation in the complex plane, Constr. Approx. 11 (1995), no. 1, 85–106. MR 1323965, DOI 10.1007/BF01294340
  • T. W. Gamelin, Uniform Algebras, Chelsea Publ. Co., New York, 1984.
  • T. W. Gamelin, Polynomial approximation on thin sets, Symposium on several complex variables (Park City, Utah, 1970) Lecture Notes in Math., Vol. 184, Springer, Berlin, 1971, pp. 50–78. MR 0300097
  • Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
  • A. B. J. Kuijlaars, The role of the endpoint in weighted polynomial approximation with varying weights, Constr. Approx. 12 (1996), 287–301.
  • A. B. J. Kuijlaars, Weighted approximation with varying weights: the case of a power type singularity, J. Math. Anal. Appl. 204 (1996), 409–418.
  • A. B. J. Kuijlaars, A note on weighted polynomial approximation with varying weights, J. Approx. Theory 87 (1996), 112–115.
  • K. Kuratowski, Topology, vol. II, Academic Press, New York, 1968.
  • G. G. Lorentz, Approximation by incomplete polynomials (problems and results), Padé and rational approximation (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976) Academic Press, New York, 1977, pp. 289–302. MR 0467089
  • D. S. Lubinsky and E. B. Saff, Uniform and mean approximation by certain weighted polynomials, with applications, Constr. Approx. 4 (1988), no. 1, 21–64. MR 916089, DOI 10.1007/BF02075447
  • Doron S. Lubinsky and Vilmos Totik, Weighted polynomial approximation with Freud weights, Constr. Approx. 10 (1994), no. 3, 301–315. MR 1291052, DOI 10.1007/BF01212563
  • I. E. Pritsker and R. S. Varga, The Szegő curve, zero distribution and weighted approximation, Trans. Amer. Math. Soc. 349 (1997), 4085–4105.
  • I. E. Pritsker and R. S. Varga, Weighted polynomial approximation in the complex plane, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 38–44.
  • Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
  • E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Heidelberg, 1997.
  • E. B. Saff and R. S. Varga, On incomplete polynomials, Numerische Methoden der Approximationstheorie, Band 4 (Meeting, Math. Forschungsinst., Oberwolfach, 1977) Internat. Schriftenreihe Numer. Math., vol. 42, Birkhäuser, Basel-Boston, Mass., 1978, pp. 281–298. MR 527107
  • N. Sibony and J. Wermer, Generators for $A(\Omega )$, Trans. Amer. Math. Soc. 194 (1974), 103–114. MR 419838, DOI 10.1090/S0002-9947-1974-0419838-9
  • Edgar Lee Stout, The theory of uniform algebras, Bogden & Quigley, Inc., Publishers, Tarrytown-on-Hudson, N.Y., 1971. MR 0423083
  • Vilmos Totik, Weighted approximation with varying weight, Lecture Notes in Mathematics, vol. 1569, Springer-Verlag, Berlin, 1994. MR 1290789, DOI 10.1007/BFb0076133
  • M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
  • John Wermer, Banach algebras and several complex variables, 2nd ed., Graduate Texts in Mathematics, No. 35, Springer-Verlag, New York-Heidelberg, 1976. MR 0394218, DOI 10.1007/978-1-4757-3878-0
Similar Articles
Additional Information
  • Igor E. Pritsker
  • Affiliation: Institute for Computational Mathematics, Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242-0001
  • Address at time of publication: Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106-7058
  • MR Author ID: 319712
  • Email:,
  • Received by editor(s): September 4, 1996
  • Received by editor(s) in revised form: March 25, 1997
  • Communicated by: Theodore W. Gamelin
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 3283-3292
  • MSC (1991): Primary 30E10; Secondary 30B60, 31A15, 41A30
  • DOI:
  • MathSciNet review: 1452821