Analytic capacity and approximation problems
Author:
A. M. Davie
Journal:
Trans. Amer. Math. Soc. 171 (1972), 409444
MSC:
Primary 30A82
MathSciNet review:
0350009
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Abstract: We consider some problems concerning analytic capacity as a set function, which are relevant to approximation problems for analytic functions on plane sets. In particular we consider the question of semiadditivity of capacity. We obtain positive results in some special cases and give applications to approximation theory. In general we establish some equivalences among various versions of the semiadditivity question and certain questions in approximation theory.
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 A. M. Davie, Bounded approximation and Dirichlet sets, J. Functional Analysis 6 (1970), 460467. MR 0275168 (43:925)
 [2]
 , Dirichlet algebras of analytic functions, J. Functional Analysis 6 (1970), 348356. MR 42 #2300. MR 0267398 (42:2300)
 [3]
 , Real annihilating measures for , J. Functional Analysis 6 (1970), 357386. MR 0275167 (43:924)
 [4]
 A. Denjoy, Sur les fonctions analytiques uniformes à singularités discontinues, C. R. Acad. Sci. Paris 149 (1909), 258260.
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 J. Garnett, Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 24 (1970), 696699. MR 0276456 (43:2203)
 [6]
 , Analytic capacity and measure (to appear).
 [7]
 T. W. Gamelin, Uniform algebras, PrenticeHall, Englewood Cliffs, N. J., 1969. MR 0410387 (53:14137)
 [8]
 T. W. Gamelin and J. Garnett, Constructive techniques in rational approximation, Trans. Amer. Math. Soc. 143 (1969), 187200. MR 40 #2882. MR 0249639 (40:2882)
 [9]
 , Pointwise bounded approximation and Dirichlet algebras, J. Functional Analysis 8 (1971), 360404. MR 0295085 (45:4153)
 [10]
 , Uniform approximation to bounded analytic functions, Rev. Un. Mat. Argentina 25 (1970), 8794. MR 0372209 (51:8425)
 [11]
 L. D. Ivanov, On Denjoy's conjecture, Uspehi Mat. Nauk 18 (1963), no. 4 (112), 147149. (Russian) MR 28 #236. MR 0156995 (28:236)
 [12]
 A. Stray, An approximation theorem for subalgebras of , Pacific J. Math. 35 (1970), 511515. MR 0276775 (43:2515)
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 M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo, 1954. MR 0114894 (22:5712)
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 A. G. Vituškin, Analytic capacity of sets in problems of approximation theory, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 141199 = Russian Math. Surveys 22 (1967), no. 6, 139200. MR 37 #5404. MR 0229838 (37:5404)
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 , Estimates of the Cauchy integral, Mat. Sb. 71 (113) (1966), 515534; English transl., Amer. Math. Soc. Transl. (2) 80 (1969), 257278. MR 34 #6124. MR 0206305 (34:6124)
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 , Examples of a set of positive length but of zero analytic capacity, Dokl. Akad. Nauk SSSR 127 (1959), 246249. (Russian) MR 22 #9607. MR 0118838 (22:9607)
 [17]
 L. Zalcman, Analytic capacity and rational approximation, Lecture Notes in Math., no. 50, SpringerVerlag, Berlin and New York, 1968. MR 37 #3018. MR 0227434 (37:3018)
 [18]
 B. Øksendsl, as a Dirichlet algebra and representation of orthogonal measures by differentials, Math. Scand. 29 (1971), 87103. MR 0306919 (46:6040)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197203500099
PII:
S 00029947(1972)03500099
Keywords:
Analytic capacity,
uniform approximation,
bounded approximation,
negligible set
Article copyright:
© Copyright 1972
American Mathematical Society
