Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Polynomial approximation with varying weights on compact sets of the complex plane

Author: Igor E. Pritsker
Journal: Proc. Amer. Math. Soc. 126 (1998), 3283-3292
MSC (1991): Primary 30E10; Secondary 30B60, 31A15, 41A30
MathSciNet review: 1452821
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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 $A(E,W).$

References [Enhancements On Off] (What's this?)

  • H. Alexander, Polynomial approximation and analytic structure, Duke Math. J. 38 (1971), 123–135. MR 283244
  • H. Alexander, Polynomial approximation and hulls in sets of finite linear measure in Cn, Amer. J. Math. 93 (1971), 65–74. MR 284617, DOI
  • Robert George Blumenthal, Holomorphically closed algebras of analytic functions, Math. Scand. 34 (1974), 84–90. MR 380423, DOI
  • P. B. Borwein and Weiyu Chen, Incomplete rational approximation in the complex plane, Constr. Approx. 11 (1995), no. 1, 85–106. MR 1323965, DOI
  • 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) Springer, Berlin, 1971, pp. 50–78. Lecture Notes in Math., Vol. 184. 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
  • Doron S. Lubinsky and Vilmos Totik, Weighted polynomial approximation with Freud weights, Constr. Approx. 10 (1994), no. 3, 301–315. MR 1291052, DOI
  • 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.
  • W. Rudin, The closed ideals in an algebra of analytic functions, Can. J. Math. 9 (1957), 426–434.
  • 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
  • Edgar Lee Stout, The theory of uniform algebras, Bogden & Quigley, Inc., 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
  • 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., Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 35. MR 0394218

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30E10, 30B60, 31A15, 41A30

Retrieve articles in all journals with MSC (1991): 30E10, 30B60, 31A15, 41A30

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

Keywords: Weighted polynomials, closed ideals, weighted energy problem, logarithmic potentials, uniform algebras
Received by editor(s): September 4, 1996
Received by editor(s) in revised form: March 25, 1997
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1998 American Mathematical Society