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

DOI:
https://doi.org/10.1090/S0002-9939-98-04402-5

MathSciNet review:
1452821

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a compact set with connected complement, let be the uniform algebra of functions continuous on and analytic interior to We describe the set of uniform limits on of sequences of the weighted polynomials as where is a nonvanishing weight on If has empty interior, then is completely characterized by a zero set However, if is a closure of Jordan domain, the description of 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

**1.**H. Alexander,*Polynomial approximation and analytic structure*, Duke Math. J.**38**(1971), 123-135. MR**44:477****2.**H. Alexander,*Polynomial approximation and hulls in sets of finite linear measure in*, Amer. J. Math.**93**(1971), 65-74. MR**44:1841****3.**R. G. Blumenthal,*Holomorphically closed algebras of analytic functions*, Math. Scand.**34**(1974), 84-90. MR**52:1323****4.**P. B. Borwein and W. Chen,*Incomplete rational approximation in the complex plane*, Constr. Approx.**11**(1995), 85-106. MR**95k:41024****5.**T. W. Gamelin, Uniform Algebras, Chelsea Publ. Co., New York, 1984.**6.**T. W. Gamelin,*Polynomial approximation on thin sets*, Symposium on Several Complex Variables (R. M. Brooks, ed.), pp. 50-78, Lecture Notes in Math., vol. 184, Springer-Verlag, Berlin, 1971. MR**45:9145****7.**K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, 1962. MR**24:A2844****8.**A. B. J. Kuijlaars,*The role of the endpoint in weighted polynomial approximation with varying weights*, Constr. Approx.**12**(1996), 287-301.**9.**A. B. J. Kuijlaars,*Weighted approximation with varying weights: the case of a power type singularity*, J. Math. Anal. Appl.**204**(1996), 409-418. CMP**97:04****10.**A. B. J. Kuijlaars,*A note on weighted polynomial approximation with varying weights*, J. Approx. Theory**87**(1996), 112-115. CMP**97:01****11.**K. Kuratowski, Topology, vol. II, Academic Press, New York, 1968.**12.**G. G. Lorentz,*Approximation by incomplete polynomials (problems and results)*, Padé and Rational Approximations: Theory and Applications (E. B. Saff and R. S. Varga, eds.), pp. 289-302, Academic Press, New York, 1977. MR**57:6956****13.**D. S. Lubinsky and E. B. Saff,*Uniform and mean approximation by certain weighted polynomials, with applications*, Constr. Approx.**4**(1988), 21-64. MR**88j:41049****14.**D. S. Lubinsky and V. Totik,*Weighted polynomial approximation with Freud weights*, Constr. Approx.**10**(1994), 301-315. MR**95i:41007****15.**I. E. Pritsker and R. S. Varga,*The Szeg\H{o} curve, zero distribution and weighted approximation*, Trans. Amer. Math. Soc.**349**(1997), 4085-4105. CMP**96:17****16.**I. E. Pritsker and R. S. Varga,*Weighted polynomial approximation in the complex plane*, Electron. Res. Announc. Amer. Math. Soc.**3**(1997), 38-44. CMP**97:11****17.**W. Rudin,*The closed ideals in an algebra of analytic functions*, Can. J. Math.**9**(1957), 426-434. MR**19:641c****18.**E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Heidelberg, 1997.**19.**E. B. Saff and R. S. Varga,*On incomplete polynomials*, Numerische Methoden der Approximationstheorie (L. Collatz, G. Meinardus and H. Werner, eds.), ISNM 42, pp. 281-298, Birkhäuser-Verlag, Basel, 1978. MR**80d:41008****20.**N. Sibony and J. Wermer,*Generators for*, Trans. Amer. Math. Soc.,**194**(1974), 103-114. MR**54:7856****21.**E. L. Stout, The Theory of Uniform Algebras, Bogden and Quigley, Inc., Belmont, 1971. MR**54:11066****22.**V. Totik, Weighted Approximation with Varying Weights, Lecture Notes in Math., vol. 1569, Springer-Verlag, Heidelberg, 1994. MR**96f:41002****23.**M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959. MR**22:5712****24.**J. Wermer, Banach Algebras and Several Complex Variables, Springer-Verlag, New York, 1976. MR**52:15021**

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

Email:
pritsker@mcs.kent.edu, iep@po.cwru.edu

DOI:
https://doi.org/10.1090/S0002-9939-98-04402-5

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