Analytic capacity and approximation problems

Davie, A. M.

doi:10.1090/S0002-9947-1972-0350009-9

Analytic capacity and approximation problems

Author: A. M. Davie
Journal: Trans. Amer. Math. Soc. 171 (1972), 409-444
MSC: Primary 30A82
DOI: https://doi.org/10.1090/S0002-9947-1972-0350009-9
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.

[1] Alexander M. Davie, Bounded approximation and Dirichlet sets, J. Functional Analysis 6 (1970), 460–467. MR 0275168, https://doi.org/10.1016/0022-1236(70)90073-x
[2] A. M. Davie, Dirichlet algebras of analytic functions, J. Functional Analysis 6 (1970), 348–356. MR 0267398, https://doi.org/10.1016/0022-1236(70)90066-2
[3] Alexander M. Davie, Real annihilating measures for 𝑅(𝐾), J. Functional Analysis 6 (1970), 357–386. MR 0275167, https://doi.org/10.1016/0022-1236(70)90067-4
[4] A. Denjoy, Sur les fonctions analytiques uniformes à singularités discontinues, C. R. Acad. Sci. Paris 149 (1909), 258-260.
[5] John Garnett, Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 24 (1970), 696-699; errata, ibid. 26 (1970), 701. MR 0276456, https://doi.org/10.1090/S0002-9939-1970-0276456-5
[6] -, Analytic capacity and measure (to appear).
[7] Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR 0410387
[8] T. W. Gamelin and J. Garnett, Constructive techniques in rational approximation, Trans. Amer. Math. Soc. 143 (1969), 187–200. MR 249639, https://doi.org/10.1090/S0002-9947-1969-0249639-4
[9] T. W. Gamelin and John Garnett, Pointwise bounded approximation and Dirichlet algebras, J. Functional Analysis 8 (1971), 360–404. MR 0295085, https://doi.org/10.1016/0022-1236(71)90002-4
[10] T. W. Gamelin and John Garnett, Uniform approximation to bounded analytic functions, Rev. Un. Mat. Argentina 25 (1970/71), 87–94. MR 372209
[11] L. D. Ivanov, On Denjoy’s conjecture, Uspehi Mat. Nauk 18 (1963), no. 4 (112), 147–149 (Russian). MR 0156995
[12] Arne Stray, An approximation theorem for subalgebras of 𝐻_{∞}, Pacific J. Math. 35 (1970), 511–515. MR 276775
[13] M. Tsuji, Potential theory in modern function theory, Maruzen Co., Ltd., Tokyo, 1959. MR 0114894
[14] A. G. Vituškin, Analytic capacity of sets in problems of approximation theory, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 141–199 (Russian). MR 0229838
[15] A. G. Vituškin, Estimate of the Cauchy integral, Mat. Sb. (N.S.) 71 (113) (1966), 515–534 (Russian). MR 0206305
[16] A. G. Vituškin, Example of a set of positive length but of zero analytic capacity, Dokl. Akad. Nauk SSSR 127 (1959), 246–249 (Russian). MR 0118838
[17] Lawrence Zalcmann, Analytic capacity and rational approximation, Lecture Notes in Mathematics, No. 50, Springer-Verlag, Berlin-New York, 1968. MR 0227434
[18] Bernt Øksendal, 𝑅(𝑋) as a Dirichlet algebra and representation of orthogonal measures by differentials, Math. Scand. 29 (1971), 87–103. MR 306919, https://doi.org/10.7146/math.scand.a-11037