×

zbMATH — the first resource for mathematics

Approximation in the mean by analytic functions. (English) Zbl 0236.30045

MSC:
30E10 Approximation in the complex plane
30H05 Spaces of bounded analytic functions of one complex variable
46E15 Banach spaces of continuous, differentiable or analytic functions
30C85 Capacity and harmonic measure in the complex plane
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Thomas Bagby, \?_{\?} approximation by analytic functions, J. Approximation Theory 5 (1972), 401 – 404. Collection of articles dedicated to J. L. Walsh on his 75th birthday, IV. · Zbl 0244.30037
[2] -, Quasi topologies and rational approximation, J. Functional Analysis (submitted). · Zbl 0266.30024
[3] Lipman Bers, An approximation theorem, J. Analyse Math. 14 (1965), 1 – 4. · Zbl 0134.05304
[4] James E. Brennan, Invariant subspaces and rational approximation, J. Functional Analysis 7 (1971), 285 – 310. · Zbl 0214.37604
[5] A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85 – 139. · Zbl 0047.10201
[6] T. Carleman, Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen, Ark. Mat. Astr. Fys. 17 (1923), 1-30. · JFM 49.0708.03
[7] Lennart Carleson, Mergelyan’s theorem on uniform polynomial approximation, Math. Scand. 15 (1964), 167 – 175. · Zbl 0163.08601
[8] Lennart Carleson, Selected problems on exceptional sets, Van Nostrand Mathematical Studies, No. 13, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. · Zbl 0189.10903
[9] Jacques Deny, Sur la convergence de certaines intégrales de la théorie du potentiel, Arch. Math. (Basel) 5 (1954), 367 – 370 (French). · Zbl 0057.33104
[10] Nicolaas du Plessis, A theorem about fractional integrals, Proc. Amer. Math. Soc. 3 (1952), 892 – 898. · Zbl 0048.03802
[11] Bent Fuglede, On generalized potentials of functions in the Lebesgue classes, Math. Scand 8 (1960), 287 – 304. · Zbl 0196.42002
[12] Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. · Zbl 0213.40401
[13] A. A. Gončar, On the uniform approximation of continuous functions by harmonic functions, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1239 – 1250 (Russian). · Zbl 0178.46202
[14] A. A. Gončar, On the approximation of continuous functions by harmonic functions, Dokl. Akad. Nauk SSSR 154 (1964), 503 – 506 (Russian).
[15] A. A. Gončar, On the property of instability of harmonic capacity, Dokl. Akad. Nauk SSSR 165 (1965), 479 – 481 (Russian).
[16] V. P. Havin, Approximation by analytic functions in the mean, Dokl. Akad. Nauk SSSR 178 (1968), 1025 – 1028 (Russian).
[17] V. G. Maz\(^{\prime}\)ja and V. P. Havin, Approximation in the mean by analytic functions, Vestnik Leningrad. Univ. 23 (1968), no. 13, 62 – 74 (Russian, with English summary).
[18] S. Ya. Havinson, Extremal problems for certain classes of analytic functions in finitely connected regions, Amer. Math. Soc. Transl. (2) 5 (1957), 1 – 33. · Zbl 0077.07802
[19] Lars Inge Hedberg, Weighted mean approximation in Carathéodory regions, Math. Scand. 23 (1968), 113 – 122 (1969). · Zbl 0182.40104
[20] Основы современной теории потенциала, Издат. ”Наука”, Мосцощ, 1966 (Руссиан).
[21] Ju. A. Lysenko and B. M. Pisarevskiĭ, The instability of harmonic capacity and the approximation of continuous functions by harmonic functions, Mat. Sb. (N.S.) 76 (118) (1968), 52 – 71 (Russian).
[22] S. N. Mergeljan, On the completeness of systems of analytic functions, Amer. Math. Soc. Transl. (2) 19 (1962), 109 – 166.
[23] S. O. Sinanjan, The uniqueness property of analytic functions on closed see without interior points, Sibirsk. Mat. Ž. 6 (1965), 1365 – 1381 (Russian).
[24] S. O. Sinanjan, Approximation by analytical functions and polynomials in the mean with respect to the area, Mat. Sb. (N.S.) 69 (111) (1966), 546 – 578 (Russian).
[25] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[26] M. Tsuji, Potential theory in modern function theory, Maruzen Co., Ltd., Tokyo, 1959. · Zbl 0087.28401
[27] A. G. Vituškin, Analytic capacity of sets in problems of approximation theory, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 141 – 199 (Russian).
[28] Hans Wallin, A connection between \?-capacity and \?^{\?}-classes of differentiable functions, Ark. Mat. 5 (1963/1965), 331 – 341 (1963/65). · Zbl 0135.32401
[29] Lawrence Zalcmann, Analytic capacity and rational approximation, Lecture Notes in Mathematics, No. 50, Springer-Verlag, Berlin-New York, 1968.
[30] William P. Ziemer, Extremal length as a capacity, Michigan Math. J. 17 (1970), 117 – 128. · Zbl 0183.39104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.