Polynomial approximation with varying weights on compact sets of the complex plane
Author:
Igor E. Pritsker
Journal:
Proc. Amer. Math. Soc. 126 (1998), 32833292
MSC (1991):
Primary 30E10; Secondary 30B60, 31A15, 41A30
MathSciNet review:
1452821
Fulltext PDF Free Access
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 0283244
(44 #477)
 2.
H.
Alexander, Polynomial approximation and hulls in sets of finite
linear measure in Cn, Amer. J. Math. 93 (1971),
65–74. MR
0284617 (44 #1841)
 3.
Robert
George Blumenthal, Holomorphically closed algebras of analytic
functions, Math. Scand. 34 (1974), 84–90. MR 0380423
(52 #1323)
 4.
P.
B. Borwein and Weiyu
Chen, Incomplete rational approximation in the complex plane,
Constr. Approx. 11 (1995), no. 1, 85–106. MR 1323965
(95k:41024), http://dx.doi.org/10.1007/BF01294340
 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 (Park City, Utah, 1970) Springer, Berlin,
1971, pp. 50–78. Lecture Notes in Math., Vol. 184. MR 0300097
(45 #9145)
 7.
Kenneth
Hoffman, Banach spaces of analytic functions, PrenticeHall
Series in Modern Analysis, PrenticeHall, Inc., Englewood Cliffs, N. J.,
1962. MR
0133008 (24 #A2844)
 8.
A. B. J. Kuijlaars, The role of the endpoint in weighted polynomial approximation with varying weights, Constr. Approx. 12 (1996), 287301.
 9.
A. B. J. Kuijlaars, Weighted approximation with varying weights: the case of a power type singularity, J. Math. Anal. Appl. 204 (1996), 409418. CMP 97:04
 10.
A. B. J. Kuijlaars, A note on weighted polynomial approximation with varying weights, J. Approx. Theory 87 (1996), 112115. 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 approximation (Proc. Internat.
Sympos., Univ. South Florida, Tampa, Fla., 1976) Academic Press, New
York, 1977, pp. 289–302. MR 0467089
(57 #6956)
 13.
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
(88j:41049), http://dx.doi.org/10.1007/BF02075447
 14.
Doron
S. Lubinsky and Vilmos
Totik, Weighted polynomial approximation with Freud weights,
Constr. Approx. 10 (1994), no. 3, 301–315. MR 1291052
(95i:41007), http://dx.doi.org/10.1007/BF01212563
 15.
I. E. Pritsker and R. S. Varga, The Szeg\H{o} curve, zero distribution and weighted approximation, Trans. Amer. Math. Soc. 349 (1997), 40854105. 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), 3844. CMP 97:11
 17.
Walter
Rudin, The closed ideals in an algebra of analytic functions,
Canad. J. Math. 9 (1957), 426–434. MR 0089254
(19,641c)
 18.
E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, SpringerVerlag, Heidelberg, 1997.
 19.
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, BaselBoston, Mass., 1978, pp. 281–298. MR 527107
(80d:41008)
 20.
N.
Sibony and J.
Wermer, Generators for
𝐴(Ω), Trans. Amer. Math. Soc.
194 (1974),
103–114. MR 0419838
(54 #7856), http://dx.doi.org/10.1090/S00029947197404198389
 21.
Edgar
Lee Stout, The theory of uniform algebras, Bogden &
Quigley, Inc., TarrytownonHudson, N. Y., 1971. MR 0423083
(54 #11066)
 22.
Vilmos
Totik, Weighted approximation with varying weight, Lecture
Notes in Mathematics, vol. 1569, SpringerVerlag, Berlin, 1994. MR 1290789
(96f:41002)
 23.
M.
Tsuji, Potential theory in modern function theory, Maruzen
Co., Ltd., Tokyo, 1959. MR 0114894
(22 #5712)
 24.
John
Wermer, Banach algebras and several complex variables, 2nd
ed., SpringerVerlag, New YorkHeidelberg, 1976. Graduate Texts in
Mathematics, No. 35. MR 0394218
(52 #15021)
 1.
 H. Alexander, Polynomial approximation and analytic structure, Duke Math. J. 38 (1971), 123135. MR 44:477
 2.
 H. Alexander, Polynomial approximation and hulls in sets of finite linear measure in , Amer. J. Math. 93 (1971), 6574. MR 44:1841
 3.
 R. G. Blumenthal, Holomorphically closed algebras of analytic functions, Math. Scand. 34 (1974), 8490. MR 52:1323
 4.
 P. B. Borwein and W. Chen, Incomplete rational approximation in the complex plane, Constr. Approx. 11 (1995), 85106. 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. 5078, Lecture Notes in Math., vol. 184, SpringerVerlag, Berlin, 1971. MR 45:9145
 7.
 K. Hoffman, Banach spaces of analytic functions, PrenticeHall, 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), 287301.
 9.
 A. B. J. Kuijlaars, Weighted approximation with varying weights: the case of a power type singularity, J. Math. Anal. Appl. 204 (1996), 409418. CMP 97:04
 10.
 A. B. J. Kuijlaars, A note on weighted polynomial approximation with varying weights, J. Approx. Theory 87 (1996), 112115. 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. 289302, 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), 2164. MR 88j:41049
 14.
 D. S. Lubinsky and V. Totik, Weighted polynomial approximation with Freud weights, Constr. Approx. 10 (1994), 301315. 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), 40854105. 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), 3844. CMP 97:11
 17.
 W. Rudin, The closed ideals in an algebra of analytic functions, Can. J. Math. 9 (1957), 426434. MR 19:641c
 18.
 E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, SpringerVerlag, 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. 281298, BirkhäuserVerlag, Basel, 1978. MR 80d:41008
 20.
 N. Sibony and J. Wermer, Generators for , Trans. Amer. Math. Soc., 194 (1974), 103114. 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, SpringerVerlag, 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, SpringerVerlag, New York, 1976. MR 52:15021
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 442420001
Address at time of publication:
Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 441067058
Email:
pritsker@mcs.kent.edu, iep@po.cwru.edu
DOI:
http://dx.doi.org/10.1090/S0002993998044025
PII:
S 00029939(98)044025
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
