zbMATH — the first resource for mathematics

Strong linear convexity. I: Duality of spaces of holomorphic functions. (English. Russian original) Zbl 0596.32017
Sib. Math. J. 26, 331-341 (1985); translation from Sib. Mat. Zh. 26, No. 3(151), 31-43 (1985).
The author studies the duality of the spaces of holomorphic functions in open (or close) subsets of the complex projective space \({\mathbb{C}}P^ n\), \(n\geq 1\). The main result is the geometric characterization of the strong linear convexity, which was introduced by A. Martineau [Anais Acad. Brasil. Ci. 40, 427-435 (1968; Zbl 0189.366)].
Reviewer: A.Zabulionis

32C37 Duality theorems for analytic spaces
32A38 Algebras of holomorphic functions of several complex variables
Full Text: DOI
[1] V. P. Khavin, ?Space of analytic functions,? in: Mathematical Analysis 1964 [in Russian], Itogi Nauki, VINITI, Moscow (1966), pp. 76-164.
[2] L. A. Rubel and B. A. Taylor, ?Functional analysis proofs of some theorems in function theory,? Am. Math. Mon., No. 5, 483-488 (1969). · Zbl 0175.44303 · doi:10.2307/2316954
[3] G. Köthe, ?Dualität in der Funktionentheorie,? J. Reine Angew. Math.,191, 30-49 (1953). · Zbl 0050.33502 · doi:10.1515/crll.1953.191.30
[4] A. Grothendieck, ?Sur certains espaces de fonctions holomorphes. I, II,? J. Reine Angew. Math.,192, No. 1, 35-64; No. 2, 77-95 (1953). · Zbl 0051.08704 · doi:10.1515/crll.1953.192.35
[5] J. Sebastiao e Silva, ?As funcoes analiticas e analyse functional,? Portugal. Math.,9, 1-130 (1950).
[6] V. P. Khavin, ?Separation of singularities of analytic functions,? Dokl. Akad. Nauk SSSR,121, No. 2, 239-248 (1958). · Zbl 0098.27702
[7] P. Levy, Concrete Problems of Functional Analysis [in Russian], Nauka, Moscow (1967).
[8] H. G. Tillman, ?Randwertverteilungen und analytischer Funktionen und Distributionen,? Math. Z.,59, 61-83 (1953). · Zbl 0051.08901 · doi:10.1007/BF01180242
[9] A. Martineau, ?Sur les equations aux derivees partielles a coefficients constants avec second membre, dans le champ complexe,? C. R. Acad. Sci.,264, 400-401 (1967). · Zbl 0143.32404
[10] A. Martineau, ?Sur la notion d’ensemble fortement lineelement convexe,? Anais Acad. Brasill. Cienc.,4, No. 4, 427-435 (1968). · Zbl 0189.36604
[11] A. Martineau, ?Equations differentielles d’ordre infini,? Bull. Soc. Math. France,95, 109-154 (1967). · Zbl 0167.44202
[12] Andre Martineau, Oeuvres, CNRE, Paris (1977).
[13] L. A. Aizenberg, ?Linear convexity inC n and separation of singularities of holomorphic functions,? Bull. Acad. Polon. Sci. Ser. Math.,15, No. 7, 487-495 (1967).
[14] L. A. Aizenberg, ?Expansion of holomorphic functions of several variables in partial fractions,? Sib. Mat. Zh.,8, No. 5, 1124-1142 (1967). · Zbl 0159.37401
[15] V. M. Trutnev, ?Analogue of Laurent series for functions of several complex variables, holomorphic on strongly linearly convex sets,? in: Holomorphic Functions of Several Complex Variables [in Russian], IF Sib. Otd. Akad. Nauk SSSR, Krasnoyarsk (1972), pp. 139-152.
[16] V. M. Trutnev, ?Properties of functions, holomorphic on strongly linear convex sets,? in: Some Properties of Holomorphic Functions of Several Complex Variables [in Russian], IF Sib. Otd. Akad. Nauk SSSR, Krasnoyarsk (1973), pp. 139-155.
[17] L. A. Aizenberg, ?General form of a continuous linear functional on spaces of functions, holomorphic in convex domains inC n,? Dokl. Akad. Nauk SSSR,166, No. 5, 1015-1018 (1966).
[18] A. P. Yuzhakov and V. P. Krivokolesko, ?Linearly convex domains with smooth boundaries inC n,? Sib. Mat. Zh.,12, No. 2, 452-458 (1971). · Zbl 0225.32006
[19] L. A. Aizenberg, ?Linear functionals on spaces of analytic functions and linear convexity inC n,? Investigations on Linear Operators and the Theory of Functions. 99 Unsolved Problems of Linear and Complex Analysis, J. Sov. Math.,26, No. 5 (1984).
[20] S. V. Znamenskii, ?Geometric tests for strong linear convexity,? Funkts. Anal.,13, No. 3, 83-84 (1979).
[21] A. Martineau, ?Sur la topologie des espaces de fonctions holomorphes,? Math. Ann.,163, No. 1, 62-88 (1966). · Zbl 0138.38101 · doi:10.1007/BF02052485
[22] B. V. Shabat, Introduction to Complex Analysis [in Russian], Part 2, Nauka, Moscow (1976).
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.