Arkiv för Matematik
- Ark. Mat.
- Volume 15, Number 1-2 (1977), 117-168.
Approximation in the mean by polynomials on non-Carathéodory domains
Full-text: Open access
Article information
Source
Ark. Mat., Volume 15, Number 1-2 (1977), 117-168.
Dates
Received: 22 September 1975
First available in Project Euclid: 31 January 2017
Permanent link to this document
https://projecteuclid.org/euclid.afm/1485896523
Digital Object Identifier
doi:10.1007/BF02386037
Mathematical Reviews number (MathSciNet)
MR450566
Zentralblatt MATH identifier
0366.30010
Rights
1977 © Institut Mittag-Leffler
Citation
Brennan, James E. Approximation in the mean by polynomials on non-Carathéodory domains. Ark. Mat. 15 (1977), no. 1-2, 117--168. doi:10.1007/BF02386037. https://projecteuclid.org/euclid.afm/1485896523
References
- Bagby, T., Quasi-topologies and rational approximation, J. Functional Analysis 10 (1972), 259–268.Zentralblatt MATH: 0266.30024
Digital Object Identifier: doi:10.1016/0022-1236(72)90025-0
Mathematical Reviews (MathSciNet): MR355058 - Bang, T., Om quasi-analytiske funktioner, Thesis, Copenhagen (1946).
- Bernštein, S. N., On weight functions, Dokl. Akad. Nauk SSSR 77 (1951), 549–552.Zentralblatt MATH: 0042.07201
- Bers, L., An approximation theorem, J. Analyse Math. 14 (1965), 1–4.Zentralblatt MATH: 0134.05304
Mathematical Reviews (MathSciNet): MR178287
Digital Object Identifier: doi:10.1007/BF02806376 - Brennan, J., Invariant subspaces and rational approximation, J. Functional Analysis 7 (1971), 285–310.Zentralblatt MATH: 0214.37604
Digital Object Identifier: doi:10.1016/0022-1236(71)90036-X
Mathematical Reviews (MathSciNet): MR423059 - Brennan, J., Invariant subspaces and weighted polynomial approximation, Ark. Mat. 11 (1973), 167–189.Zentralblatt MATH: 0272.46022
Digital Object Identifier: doi:10.1007/BF02388514
Mathematical Reviews (MathSciNet): MR350398 - Brennan, J., Approximation in the mean and quasi-analyticity, J. Functional Analysis 12 (1973), 307–320.Zentralblatt MATH: 0263.30031
Digital Object Identifier: doi:10.1016/0022-1236(73)90082-7
Mathematical Reviews (MathSciNet): MR344481 - Calderón, A. P., Zygmund, A., On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139.Zentralblatt MATH: 0047.10201
Digital Object Identifier: doi:10.1007/BF02392130
Mathematical Reviews (MathSciNet): MR52553 - Carleman, T., Über die approximation analytischer funktionen durch lineare aggregate von vorgegebenen potenzen, Ark. Mat. Astr. Fys. 17 (1923), 1–30.
- Carleson, L., Interpolations by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547–559.
- Carleson, L., Mergeljan's theorem on uniform polynomial approximation, Math. Scand. 15 (1965), 167–175.Mathematical Reviews (MathSciNet): MR198209
- Carleson, L., Jacobs, S., Best uniform approximation by analytic functions, Ark. Mat. 10 (1972), 219–229.Zentralblatt MATH: 0248.30034
Digital Object Identifier: doi:10.1007/BF02384810
Mathematical Reviews (MathSciNet): MR322410 - Dales, H. G., and Davie, A. M., Quasianalytic Banach function algebras, J. Functional Analysis 13 (1973), 28–50.Zentralblatt MATH: 0254.46027
Digital Object Identifier: doi:10.1016/0022-1236(73)90065-7
Mathematical Reviews (MathSciNet): MR343038 - Deny, J., Systèmes totaux de fonctions harmoniques, Ann. Inst. Fourier 1 (1949), 103–113.Mathematical Reviews (MathSciNet): MR37414
- Duren, P. L., Theory of Hpspaces, Academic Press, New York (1970).Zentralblatt MATH: 0215.20203
- Dyn'kin, E. M., Functions with a given estimate for
$\partial f/\partial \bar z$
and N. Levinson's theorem, Math. U.S.S.R. Sbornik 18 (1972), 181–189, Mat. Sb. 89 (1972), 182–190.Digital Object Identifier: doi:10.1070/SM1972v018n02ABEH001753
- Džrbašjan, M. M., Metric theory of completeness and representation of analytic functions, Thesis, Erevan (1948).
- Farrell, O. J., On approximation to an analytic function by polynomials, Amer. Math. Soc. Bull. 40 (1934), 908–914.Digital Object Identifier: doi:10.1090/S0002-9904-1934-06002-6
Mathematical Reviews (MathSciNet): MR1562998 - Federer, H., Curvature measures, Amer. Math. Soc. Trans. 93 (1959), 418–491.Zentralblatt MATH: 0089.38402
Digital Object Identifier: doi:10.2307/1993504
Mathematical Reviews (MathSciNet): MR110078 - Federer, H., Geometric measure theory, Springer-Verlag, Berlin (1969).Zentralblatt MATH: 0176.00801
- Hardy, G. H., Littlewood, J. E., and Pólya, G., Inequalities, Cambridge University Press (1964).
- Havin, V. P., Approximation in the mean by analytic functions, Soviet Math. Dokl. 9 (1968), 245–248, Dokl. Akad. Nauk SSSR 178 (1968), 1025–1028.
- Havin, V. P., Approximation in the mean by polynomials on certain non-Carathéodory domains I & II, Izv. Vysš. Učebn. Zaved Mat. 76, (1968), 86–93 & 77 (1968), 87–94.Mathematical Reviews (MathSciNet): MR237796
- Havin, V. P. A generalization of the Privalov-Zygmund theorem on the modulus of continuity of the conjugate function, Izv. Akad. Nauk Armjan SSR 6 (1971), 252–258 & 265–287.Mathematical Reviews (MathSciNet): MR302923
- Havin, V. P., and Maz'ja, V. G., Approximation in the mean by analytic functions, Vestnik Leningrad Univ. 13 (1968), 64–74.
- Havin, V. P. and Maz'ja, V. G., Approximation in the mean by harmonic functions, Zap. Naučn. Sem. Leningrad Otdel. Mat. Inst. Steklov (LOMI)30 (1972), 91–105.Mathematical Reviews (MathSciNet): MR328107
- Havin, V. P. and Maz'ja, V. G., Non-linear potential theory, Russian Math. Surveys 27 (1972), 71–148, Uspehi Mat. Nauk 27 (1972), 67–138.Mathematical Reviews (MathSciNet): MR409858
- Havin, V. P. and Maz'ja, V. G., Applications of (p, l)-capacity to some problems in the theory of exceptional sets, Math. USSR Sbornik 19 (1973), 547–580, Mat. Sb. 90 (1973), 558–591.Digital Object Identifier: doi:10.1070/SM1973v019n04ABEH001808
- Havin, V. P. and Maz'ja, V. G., On solutions of the Cauchy problem for the Laplace equation (uniqueness, normality, approximation), Trudy Moskov. Mat. Obšč. 30 (1974), 61–114.Mathematical Reviews (MathSciNet): MR385135
- Hedberg, L. I., Weighted mean square approximation in plane regions and generators of an algebra of analytic functions, Ark. Mat. 5 (1965), 541–552.Zentralblatt MATH: 0154.38701
Digital Object Identifier: doi:10.1007/BF02591530
Mathematical Reviews (MathSciNet): MR219729 - Hedberg, L. I., Weighted mean approximation in Carathéodory regions, Math. Scand. 23 (1968), 113–122.
- Hedberg, L. I., Approximation in the mean by analytic functions, Amer. Math. Soc. Trans. 163 (1972), 157–171.Zentralblatt MATH: 0236.30045
Digital Object Identifier: doi:10.2307/1995714
Mathematical Reviews (MathSciNet): MR432886 - Hedberg, L. I., Non-linear potentials and approximation in the mean by analytic functions, Math. Z. 129 (1972), 299–319.Zentralblatt MATH: 0236.31010
Digital Object Identifier: doi:10.1007/BF01181619
Mathematical Reviews (MathSciNet): MR328088 - Hedberg, L. I., Approximation in the mean by solutions of elliptic equations, Duke Math. J. 40 (1973), 9–16.Zentralblatt MATH: 0283.35035
Digital Object Identifier: doi:10.1215/S0012-7094-73-04002-7
Mathematical Reviews (MathSciNet): MR312071 - Jacobs, S., Ett extremalproblem för analytiska funktioner i multipelt sammanhängande områden, Uppsala Univ. Math. Dept. report 2 (1969).
- Keldyš, M. V., Sur l'approximation en moyenne par polynomes des fonctions d'une variable complexe, Mat. Sb. 16 (1945), 1–20.
- Landkof, N. S., Foundations of modern potential theory, Springer-Verlag, Berlin (1972).Zentralblatt MATH: 0253.31001
- Mandelbrojt, S., Séries adhérentes, régularisation des suites, applications, Gauthier-Villars, Paris (1952).Zentralblatt MATH: 0048.05203
- Markuševič, A. I., Conformal mapping of regions with variable boundary and application to the approximation of analytic functions by polynomials, Thesis, Moscow (1934).
- Mergeljan, S. N., On the completeness of systems of analytic functions, Amer. Math. Soc. Translations 19 (1962), 109–166, Uspehi Mat. Nauk 8 (1953), 3–63.Mathematical Reviews (MathSciNet): MR131561
- Mergeljan, S. N., Weighted approximation by polynomials, Amer. Math. Soc. Translations 10 (1958), 59–106, Uspehi Mat. NaukII (1956), 107–152.Mathematical Reviews (MathSciNet): MR94633
- Mergeljan, S. N., General metric criteria for the completeness of systems of polynomials, Dokl. Akad. Nauk SSSR 105 (1955), 901–904.Zentralblatt MATH: 0065.30405
- Mergeljan, S. N., Harmonic approximation and an approximate solution of the Cauchy problem for the Laplace equation, Uspehi Mat. Nauk 11 (1956), 3–26.
- Mergeljan, S. N. and Tamadjan, A. P., On completeness in a class of non-Jordan regions, Amer. Math. Soc. Translations 35 (1964), 79–94, Izv. Akad. Nauk Armjan. SSR 7 (1954), 1–17.
- Nevanlinna, R., Analytic functions, Springer-Verlag, Berlin (1970).Zentralblatt MATH: 0199.12501
- Polking, J. C., Approximation in Lp by solutions of elliptic partial differential equations, Amer. J. Math. 94 (1972), 1231–1244.Zentralblatt MATH: 0266.35026
Digital Object Identifier: doi:10.2307/2373572
Mathematical Reviews (MathSciNet): MR324215 - Rao, V., A uniqueness theorem for harmonic functions, Math. Notes 3 (1968), 159–162, Mat. Zametki 3 (1968), 547–552.Zentralblatt MATH: 0165.44902
- Šaginjan, A. L., On a criterion for the incompleteness of a system of analytic functions, Dokl. Akad. Nauk Armjan. SSR 5 (1946), 97–100.
- Šaginjan, A. L., A problem in the theory of approximation in the complex domain, Sibirsk Mat. Ž. 1 (1960), 523–543.Mathematical Reviews (MathSciNet): MR123724
- Shapiro, H. S., Some remarks on weighted polynomial approximations of holomorphic functions, Math. USSR Sbornik 2 (1967), 285–293, Mat. Sb. 73 (1967), 320–330.
- Sinanjan, S. O., Approximation by polynomials and analytic functions in the areal mean, Amer. Math. Soc. Translations 74 (1968), 91–124, Mat. Sb. 69 (1966), 546–578.
- Sinanjan, S. O., Approximation by polynomials in the mean with respect to area, Math. USSR Sbornik 11 (1970), 411–421, Mat. Sb. 82 (1970), 444–455.Digital Object Identifier: doi:10.1070/SM1970v011n03ABEH001299
- Smirnov, V. I. and Lebedev, N. A., Functions of a complex variable—constructive theory, M.I.T. Press, Cambridge, Mass. (1968).Zentralblatt MATH: 0164.37503
- Tamadjan, A. P. A theorem of M. V. Keldyš, Izv. Akad. Nauk Armjan. SSR 6 (1953), 5–11.
- Warschawski, S. E., Über einen satz von O. D. Kellogg, Göttinger Nachrichten (1932), 73–86.
- Warschawski, S. E., On conformal mapping of regions bounded by smooth curves, Amer. Math. Soc. Proc. 2 (1951), 254–261.Zentralblatt MATH: 0043.08201
Digital Object Identifier: doi:10.2307/2032496
Mathematical Reviews (MathSciNet): MR41231 - Whitehead, J. H. C., Manifolds with transverse fields in Euclidean space, Ann. of Math. 73 (1961), 154–212.
- Zygmund, A., Trigonometric series, Vol. 1, Cambridge Univ. Press (1959).
- Beurling, A., Analytic continuation across a linear boundary, Acta Math. 128 (1972), 153–182.Zentralblatt MATH: 0235.30003
Digital Object Identifier: doi:10.1007/BF02392164
Mathematical Reviews (MathSciNet): MR393559 - Beurling, A., Quasianalyticity and general distributions, Lecture notes, Stanford University (1961).
- Meyers, N. G., A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand. 26 (1970), 255–292.
- Melnikov, M. S. and Sinanjan, S. O., Questions in the theory of approximation of functions of one complex variable, Contemporary Problems of Mathematics vol. 4, Itogi Nauki i Tekhniki, VINITI, Moscow (1975), 143–250.
- Lisin, F. S., A connection between approximation of harmonic and analytic functions in the areal mean, Math. Notes 10 (1971), 807–811; Mat. Zametki 10 (1971), 627–634.Mathematical Reviews (MathSciNet): MR291473
- Havin, V. P., On approximation in Lp by solutions of certain systems of linear differential equations, Vestnik Leningrad Univ. no. 1 (1975), 150–158.Mathematical Reviews (MathSciNet): MR390752

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