Skip to main content
Log in

The contemporary development in M. M. Djrbashian factorization theory and related problems of analysis

  • Complex Analysis
  • Published:
Journal of Contemporary Mathematical Analysis Aims and scope Submit manuscript

Abstract

This survey gives the concept, the historical background and the results which lay ground for the investigations in the field of M. M. Djrbashian factorization theory and related problems of analysis. It gives some applications and results which reveal the fields of further, contemporary investigations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Biberbach, “Zur Theorie und Praxis der KonformenAbbildung”, Palermo Rendiconti, 38, 98–118 (1914).

    Article  Google Scholar 

  2. T. Carleman, “Über die Approximation Analytischer Funktionen durch Lineare Aggregate von Vorgegebenen Potenzen”, Arkhiv för Mat., Astr. och Fys., 17(9) (1922).

  3. F. Nevanlinna, R. Nevanlinna, “Über die Eigenschaften analytischer Funktionen in der Umgebung einer singulären Stelle oder Linie”, Acta Soc. Sci. Fennicae, 50(5), (1922).

  4. R. Nevanlinna, “Asymptotische Entwicklungen beschränkter Functionen und das Stieltjessche Momentenproblem”, Ann. Acad. Sci. Fenn. Ser. AI, 8(5), (1922).

  5. R. Nevanlinna, “Über die Eigenschaften meromorpher Funktionen in einem Winkelraum”, Acta Soc. Sci. Fennicae, 50(12), 3–45 (1925).

    Google Scholar 

  6. S. Takenaka, “On the Orthogonal Functions and a New Formula of Interpolation”, Japanese J. Math., 2, 129–145 (1925).

    Google Scholar 

  7. W. Wirtinger, “Über eine Minimumaufgabe im Gebiet der analytischen Functionen’, Monatshefte für Math. und Phys., 39, 377–384 (1932).

    Article  MathSciNet  Google Scholar 

  8. S. Bergman, “Uber unendliche Hermitische Formen, die zu einem Bereiche gehoren, Nebst Anwendungen auf Fragen der Abbildung durch Funktionen von Zwei Komplexen Veranderlichen”, Math. Z., 29, 641–677 (1929).

    Article  MathSciNet  Google Scholar 

  9. S. Bergman, “Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande”, I, J. Reine. Agnew. Math., 169, 1–42 (1933), II, Ibid., 172, 89–128 (1934).

    Google Scholar 

  10. G. H. Hardy, J. E. Littlewood, “Some Properties of Functional Integrals II”, Mathematische Zeitschrift, 34, 403–439 (1932).

    Article  MathSciNet  Google Scholar 

  11. R. Nevanlinna, Eindeutige Analytische Funktionen (Springer, Berlin, 1936).

    Google Scholar 

  12. M. V. Keldysch, “Sur les Conditions pour Qu’un Systeme de Polynomes Orthogonaux Avec un Poids Soit Ferme”, Coptes Rendus (Doklady) Academie des Sciences USSR (ns), 30, 778–780 (1941).

    Google Scholar 

  13. L. V. Ahlfors, M. Heins, “Questions of Regularity Connected with the Phragmen-Lindelöf Principle”, Ann. of Math., 50(2), 341–346 (1949).

    Article  MathSciNet  Google Scholar 

  14. J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Soc. Coll. Publ. XX (Edwards Brothers, Inc., Ann Arbor, Michigan, 1956).

    Google Scholar 

  15. M. M. Djrbashian, “On Canonical Representation of Functions Meromorphic in the Unit Disc”, Dokl. Akad. Nauk. Arm. SSR, 3(1), 3–9 (1945).

    Google Scholar 

  16. M. M. Djrbashian, “On the Representability Problem of Analytic Functions”, Soobsch. Inst. Matem. i Mekh. Akad. Nauk Arm. SSR, 2, 3–40 (1948).

    Google Scholar 

  17. M. M. Djrbashian, Integral Transforms and Representations of Functions in the Complex Domain (Nauka, Moscow, 1966).

    Google Scholar 

  18. M. M. Djrbashian, “Generalized Riemann-Liouville Operator and Some Applications”, Izv. Akad. Nauk SSSR, Ser. Mat., 32, 1075–1111 (1968).

    MathSciNet  Google Scholar 

  19. M. M. Djrbashian, “Theory of Factorization of Functions Meromorphic in the Unit Disc”, Math. USSR Sbornik, 8(4), 493–591 (1969).

    Article  Google Scholar 

  20. M. M. Djrbashian, “Factorization of Functions Meromorphic in the Finite Plane”, Izv. Akad. Nauk Arm. SSR, Math., 5(6), 453–485 (1970).

    Google Scholar 

  21. M. M. Djrbashian, “Biorthogonal Systems of Rational Functions and Representations by Cauchy Kernel”, Izv. AN Arm. SSR, Matematika [Soviet Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)], 8(5), 384–409 (1973).

    Google Scholar 

  22. M. M. Djrbashian, “Biorthogonal Systems and Solution of Interpolation Problem with Knots of Bounded Multiplicity in H 2”, Izv. AN Arm. SSR, Matematika [Soviet Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)], 9(5), 339–373 (1974).

    Google Scholar 

  23. M. M. Djrbashian, “Theory of Factorization and Boundary Properties of Functions Meromorphic in the Disc”, in: Proceedings of the ICM, Vancouver, B.C., 1974, 2, 197–202 (USA, 1975).

    Google Scholar 

  24. M. M. Djrbashian, “On an Infinite Product”, Dokl. Akad. Nauk SSSR, 240(3), 518–521 (1978) (Russian).

    MathSciNet  Google Scholar 

  25. M. M. Djrbashian, “Some Open Questions of the Representation Theory of Analytic Functions”, Zametki Nauch. Sem. LOMI, 81, 238–241 (1978).

    Google Scholar 

  26. M. M. Djrbashian, “Some Open Problems in the Theory of Representations of Analytic Functions”, in: Lecture Notes in Mathematics, 1043, 522–526 (Springer-Verlag, 1984).

    Google Scholar 

  27. M. M. Djrbashian, “Brief Survey of Results of Mathematicians of Armenia in on Factorization Theory of Meromorphic Functions and its Applications”, Izv. AN Arm. SSR, Matematika, [Soviet Journal of ContemporaryMathematical Analysis (Academy of Sciences of Armenia)], 22(6), 517–545 (1988).

    Google Scholar 

  28. M. M. Djrbashian, “Weighted Integral Representations of Analytic and Smooth Functions in the Unit Disc and in C”, Izv. NAN Armenii, Matematika [Journal of Contemporary Mathematical Analysis (Academy of Sciences of Armenia)], 28(4), 3–33 (1993).

    Google Scholar 

  29. M. M. Djrbashian, V. S. Zakaryan, “Boundary Properties of Meromorphic Functions from the N α Class”, Izv. AN Arm. SSR, Matematika, 2(5), 275–294 (1967).

    Google Scholar 

  30. M. M. Djrbashian, V. S. Zakaryan, “On Factorization of the Function B α”, Mat. Zametki, 4(1), 3–10 (1968).

    MathSciNet  Google Scholar 

  31. M. M. Djrbashian, V. S. Zakaryan, “On Boundary Properties of Meromorphic Functions of the N α Class”, Dokl. AN SSSR, 173(121), 1247–1250 (1969).

    Google Scholar 

  32. M. M. Djrbashian, V. S. Zakaryan, “Boundary Properties of Subclasses of Meromorphic Functions of Bounded Type”, Izv. Akad. Nauk SSSR, ser. Mat. bf 34, 1262–1339 (1970) = Math. USSR Izv., bf 4 (1970), 1273–1354.

    Google Scholar 

  33. M. M. Djrbashian, V. S. Zakaryan, “Boundary Properties of Subclasses of Meromorphic Functions of Bounded Type”, Izv. Akad. Nauk Arm. SSR, Mat. bf 6 (1971), no. 2–3, 182–194 (in Russian).

    Google Scholar 

  34. M. M. Djrbashian, V. S. Zakaryan, Classes and Boundary Properties of Functions Meromorphic in the Disc (Nauka, Moscow, 1993). (Russian)

    Google Scholar 

  35. M. M. Djrbashian, V. M. Martirosyan, “Integral Representations of Some Classes of Functions in the Strip or in the Halfplane”, Dokl. Akad. Nauk SSSR, 283(5), 1054–1057 (1985).

    MathSciNet  Google Scholar 

  36. M. M. Djrbashian, A. E. Djrbashian, “Integral Representation for Some Classes of Analytic Functions in the Halfplane”, Dokl. Akad. Nauk SSSR, 285(3), 547–550 (1985) (Russian).

    MathSciNet  Google Scholar 

  37. M. M. Djrbashian, A. H. Karapetyan, “Integral Representations of Some Classes of Functions Analytic in the Siegel Domain”, Izv. AN Arm. SSR, Matematika [Soviet Journal of Contemporary Mathematical Analysis (Academy of Sciences of Armenia)], 22(4), 399–405 (1987).

    Google Scholar 

  38. M. M. Djrbashian, A. H. Karapetyan, “Integral Representations in the Generalized Unit Disc”, Izv. AN Arm. SSR, Matematika, [Soviet Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 24(6), 1–25 (1989).

    Google Scholar 

  39. M. M. Djrbashian, A. H. Karapetyan, “Integral Representations in the GeneralizedUnit Disc”, Dokl. Akad. Nauk SSSR, 312(1), 24–27 (1990).

    Google Scholar 

  40. M. M. Djrbashian, A. H. Karapetyan, “Integral Representations in the Generalized Upper Half-Plane, Dokl. Akad. Nauk SSSR, 312(2), 270–273 (1990).

    Google Scholar 

  41. M. M. Djrbashian, A. H. Karapetyan, “Integral Representations in the Generalized Upper Half-Plane, Izv. AN Armenii, Matematika [Journal of Contemporary Mathematical Analysis (Academy of Sciences of Armenia)], 25(6), 507–533 (1990).

    Google Scholar 

  42. M. M. Djrbashian, A. H. Karapetyan, “Integral Representations and Uniqueness Theorems for Entire Functions of Several Variables”, Izv. AN Armenii, Matematika [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 26(1), 3–19 (1991).

    Google Scholar 

  43. M. M. Djrbashian, A. H. Karapetyan, “Integral Representations for Some Classes of Functions Holomorphic in a Siegel domain”, J. Math. Anal. Appl., 179(1), 91–109 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  44. V. S. Zakaryan, “On Radial Limiting Values of a Class of Functions Meromorphic in the Disc”, Izv. AN SSSR, Ser. Mat., 27(4), 801–818 (1963). (Russian)

    Google Scholar 

  45. V. S. Zakaryan, “Uniqueness Theorems for some Classes of Functions Meromorphic in the Disc”, Dokl. AN Arm. SSR, 36(1), 3–9 (1963). (Russian)

    Google Scholar 

  46. V. S. Zakaryan, “Uniqueness Theorems for some Classes of Functions Holomorphic in the Disc”, Mat. Sbornik, 10(1), 3–22 (1963). (Russian)

    Google Scholar 

  47. V. S. Zakaryan, “On Segmental Variation of a Class of Analytic Functions”, Izw. AN Arm. SSR, Matematika, 2(2), 117–122 (1967). (Russian)

    Google Scholar 

  48. V. S. Zakaryan, “Radial Limits and Radial Variations of the Product B α”, Izw. AN Arm. SSR, Matematika, 3(1), 38–51 (1968). (Russian)

    Google Scholar 

  49. V. S. Zakaryan, “On Radial Limit Values of the Function B α”, Izv. AN Arm. SSR, Matematika, 3(4, 5), 287–300 (1968). (Russian)

    Google Scholar 

  50. V. S. Zakaryan, “Growth Estimate for Meromorphic Functions of the Class N α”, Izv. AN Arm. SSR, Matematika [Soviet Journal of Contemporary Mathematical Analysis (Academy of Sciences of Armenia)], 9(2), 85–106 (1974).

    Google Scholar 

  51. V. S. Zakaryan, “Segmental Variations of the Green Type Potential”, Dokl. AN Arm. SSR, 66(4), 212–215 (1978). (Russian)

    Google Scholar 

  52. V. S. Zakaryan, “On a Growth Estimate for M. M. Djrbashian Product”, Izv. AN Arm. SSR, Matematika [Soviet Journal of Contemporary Mathematical Analysis (Academy of Sciences of Armenia)], 23(2), 189–192 (1988).

    Google Scholar 

  53. V. S. Zakaryan, L. A. Sekhposyan, “A Growth Estimate for Meromorphic Functions of N“ω” Class”, Izv. AN Arm. SSR, Matematika [Soviet Journal of Contemporary Mathematical Analysis (Academy of Sciences of Armenia)], 15(5), 348–368 (1979).

    Google Scholar 

  54. V. S. Zakaryan, A. G. Hunanyan, “The Variations of the Green Type Potential on Segments”, Dokl. AN Arm. SSR, 73(1), 3–8 (1981). (Russian)

    Google Scholar 

  55. V. S. Zakaryan, S. V. Madoyan, “On Boundary Values of Functions from the Class N α (−1 < α < 0)”, Dokl. AN Arm. SSR, 39(1), 348–368 (1984). (Russian)

    Google Scholar 

  56. F. A. Shamoyan, “M. M. Djrbashian Factorization Theorem and Characterization of Zeros of Analytic in the Disc Functions With Majorant of Finite Growth”, Izv. AN Arm. SSR, Matematika [Soviet Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)], 13(5–6), 405–422 (1978).

    MATH  MathSciNet  Google Scholar 

  57. F. A. Shamoyan, “Division Theorems and Closed Ideals in Algebras of Analytic functions with Majorant of Finite Growth”, Izv. AN Arm. SSR, Matematika [Soviet Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)], 15(4), 323–332 (1980).

    MATH  MathSciNet  Google Scholar 

  58. F. A. Shamoyan, “Application of M. M. Djrbashian Integral representations to Some Analysis Problems”, Dokl. Akad. Nauk SSSR, 261(3), 557–561 (1981) (Russian).

    MathSciNet  Google Scholar 

  59. F. A. Shamoyan, “On Zeros of Analytic in the Disc Functions Growing Near the Boundary”, Izv. AN Arm. SSR, Matematika, [Soviet Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)], 18(1), 15–27 (1983).

    MathSciNet  Google Scholar 

  60. F. A. Shamoyan, “Diagonal Mapping and Representation Questions in Anisotropic Spaces of Functions Holomorphic in the Polydisc”, Dokl. Akad. Nauk. Armenii, 85(1), 7–11 (1987).

    MATH  MathSciNet  Google Scholar 

  61. A. E. Djrbashian, F. A. Shamoian, Topics in the Theory of A p α Spaces (Teubner-Texte zur Math., 105, Leipzig, 1988).

  62. F. A. Shamoyan, “Diagonal Mapping and Representation Questions in Anisotropic Spaces of Functions Holomorphic in the Polydisc”, Sib. Mat. J., 31(3), 197–216 (1990).

    MathSciNet  Google Scholar 

  63. F. A. Shamoyan, A. V. Harutyunyan, “Toeplitz Operators in Anisotropic Spaces of Functions Holomorphic in the Polydisc”, Dokladi, Akad. Nauk Armenii, 91(4), 147–151 (1990).

    Google Scholar 

  64. F. A. Shamoyan, A. V. Harutyunyan, “On Anisotropic Spaces of Holomorphic in the Polydisc Functions, Smooth up to the Boundary”, Izv. NAN Armenii, Matematika, [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 28(6), 50–64 (1993).

    MATH  Google Scholar 

  65. F. A. Shamoyan, “Some Remarks on the Parametric Representation of Nevanlinna-Djrbashian Classes”, Math. Notes, 52(1–2), 727–737 (1993).

    MathSciNet  Google Scholar 

  66. F. A. Shamoyan, A. V. Harutyunyan, “Toeplitz Operators in Multidimensional Spaces of M. M. Djrbashian”, Izv. NAN Armenii, Matematika, [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 30(2), 70–78 (1995).

    MATH  Google Scholar 

  67. F. A. Shamoyan, I. S. Kursina, “On Invariance of Some Classes of Functions Holomorphic with Respect to Integro-Differential Operators”, Zap. Nauch. Sem. POMI, 255 184–197 (1998).

    Google Scholar 

  68. F. A. Shamoyan, O. V. Yaroslavtseva, “Continuous Projectors, Duality, Diagonal Mappings in Weighted Spaces of Holomorphic Functions With Mixed Norm”, Zap. nauch. Sem., POMI, 225, 182–199 (1998).

    Google Scholar 

  69. I. S. Kursina, “On Parametric Representation of Some Classes of Functions Holomorphic in the Half-Plane”, Dep. VINITI, 30.03.98-No919-1398. (Russian)

  70. F. A. Shamoyan, “Parametric Representation and Description of the Root Sets of Classes of Functions Holomorphic in the Disc”, Sib. Mat. J., 40(6), 1420–1440 (1999).

    MathSciNet  Google Scholar 

  71. F. A. Shamoyan, E. N. Shubabko, Parametrical Representation of Some Classes of Holomorphic Functions in the Disc, in: Operator Theory, Advances and Applications, 113, 332–339 (Birkhäuser Verlag, Basel, 2000).

    Google Scholar 

  72. F. A. Shamoyan, E. N. Shubabko, “On a Class of Functions Holomorphic in the Disc”, in: Investigations on Linear Operators and Function Theory, 29, 242–256, POMI (St. Petersburg, 2001).

    Google Scholar 

  73. F. A. Shamoyan, E. N. Shubabko, “Parametric Representation of a Class of Functions Holomorphic in the Disc”, Integralnie Pred. i Spec. Funkcii, 3(1), 132–142 (2002) (Russian).

    Google Scholar 

  74. F. A. Shamoyan, N. A. Chasova, Bounded Projectors and Continuous Linear Functionals in Weighted Mixed Norm Spaces of Analytic in Polydisc Functions in the Polydisc (Bryansk State Univ. Preprint, Bryansk, 2002).

    Google Scholar 

  75. N. A. Chasova, “On Toeplitz Operators in Mixed Norm Spaces of Functions Analytic in the Polydisc”, in: TrudiMat. Centra im. N. I. Lobachevskogo, 14, 301–319 (Kazan, 2002).

    MathSciNet  Google Scholar 

  76. F. A. Shamoyan, V. A. Bednaj, “Description of Main Parts in Laurent Expansion of Some Classes of FunctionsMeromorphic in the Disc”, Vestnik Bryanskogo Univ., 4, 84–96 (2004).

    Google Scholar 

  77. O. E. Antonenkova, F. A. Shamoyan, “Description of Continuous Linear Functionals in Weighted Mixed Norm Spaces of Functions Analytic in the Polydisc”, Uspekhi Mat. Nauk., 60(4), 217–219 (2005).

    MathSciNet  Google Scholar 

  78. N. M. Tkachenko, F. A. Shamoyan. The Hardy-Littlewood Theorem and the Harmonic Conjugation Operator in Some classes of Simply Connected Domains with Rectifiable Boundary”, J. of Math. Phys., Analysis, Geometry, 5(2), pp. 1–19 (2009).

    MathSciNet  Google Scholar 

  79. H. M. Hayrapetyan, “On Basicity of Some Biorthogonal Systems in the Complex Domain”, Izv. Ac. of Sci. of Arm. SSR, Matematika [Soviet Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)], 10(2), 133–152 (1975).

    Google Scholar 

  80. D. T. Baghdasaryan, I. V. Ohanyan, “Boundary Properties of M. M. Djrbashian Functions B α(z, z k)”, Dokl. AN Arm. SSR, 90(5), 199–205 (1990) (Russian).

    Google Scholar 

  81. A. M. Jerbashian, G. V. Mikaelyan, “Construction and Main Properties of a Family of Blaschke Type Functions for the Half-Plane”, Izv. Ac. of Sci. of Arm. SSR, Matematika, [Soviet Journal of Contemporary Mathematical Analysis (Academy of Sciences of Armenia)], 15(6), 461–474 (1980)

    Google Scholar 

  82. G. M. Gubreev, A. M. Jerbashian, “Functions ofGeneralized Bounded Type in Spectral Theory ofNon-weak Contractions”, J. of Operator Theory, 26, 155–190 (1991).

    MATH  MathSciNet  Google Scholar 

  83. A. M. Jerbashian, “On Boundary Properties of Functions of Generalized Bounded Type”, Izv. National Ac. of Sci. of Armenia, Matematika, [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 26(3), 187–209 (1991).

    Google Scholar 

  84. A. M. Jerbashian, G. V. Mikayelyan, “On Boundary Properties of Blaschke Type Products”, Izv. National Ac. of Sci. of Armenia, Matematika, [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 26(5), 435–442 (1991).

    Google Scholar 

  85. A. M. Jerbashian, “On Functions of Bounded and Generalized Bounded Type in the Half-Plane”, Rev. Roum. Math. Pur. et Appl., 36(7–8), 425–432 (1991).

    MATH  MathSciNet  Google Scholar 

  86. A. M. Jerbashian, “On Subharmonic functions Having Nonnegative Harmonic Majorants in the Halfplane”, Izv. Ac. of Sci. of Armenia, Matematika [Journal of Contemporary Mathematical Analysis (Armenia National Academy of Sciences)], 28(4), 42–61 (1993).

    Google Scholar 

  87. A. M. Jerbashian, “An Extension of the Factorization Theory of M. M. Djrbashian”, Izv. National Ac. of Sci. of Armenia, Matematika [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)] 30(2), 39–61 (1995).

    MATH  Google Scholar 

  88. A. M. Jerbashian, “On Embedding of Nevanlinna Type ClassesN ω”, Izv. Ac. of Sci. of Russia, 63(4), 59–78 (1999).

    Google Scholar 

  89. A. M. Jerbashian, “Evaluation of M. M. Djrbashian ω-kernels”, Archives of Inequalities and Applications, 1, 399–422 (2003).

    MATH  MathSciNet  Google Scholar 

  90. A. M. Jerbashian, Functions of α-Bounded Type in the Half-Plane, Advances in Complex Analysis and Applications (Springer, 2005).

  91. A. M. Jerbashian, “On the Theory of Weighted Classes of Area Integrable Regular Functions”, Complex Variables, 50, 155–183 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  92. A. M. Jerbashian, “On A pω,γ Spaces in the Half-Plane”, in: Operator Theory: Advances and Applications, 158, 141–158 (Birkhäuser Verlag, Basel/Switzerland, 2005).

    Google Scholar 

  93. A. M. Jerbashian, V. A. Jerbashian, “Functions of ω-Bounded Type in the Half-Plane” CMFT: Calculation Methods and Function Theory, 7(2), 205–238 (2007).

    MATH  MathSciNet  Google Scholar 

  94. S. G. Rafayelyan, A. M. Jerbashian, “On Unitary Operators in Weighted Spaces A 2ω (ℂ) of Entire Functions”, Methods of Funct. Anal. and Topology, 14(4), 380–385 (2008).

    MATH  MathSciNet  Google Scholar 

  95. A. M. Jerbashian, “Orthogonal Decomposition of Functions Subharmonic in the Unit Disc”, in: Operator Theory: Advances and Applications, 190, The Mark Krein Centenary Conference, 1: Operator Theory and Related Topics, 335–340 (Birkhäuser, 2009).

    Google Scholar 

  96. A. H. Karapetyan, Some Questions of Integral Representations in Multidimensional Complex Analysis, Candidate Thesis, (Yerevan, 1987).

  97. A. H. Karapetyan, “Integral Representations in Tube Domains”, Izv. Ac. of Sci. of Arm. SSR, Matematika [Soviet Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)], 23(1), 91–96 (1988).

    MathSciNet  Google Scholar 

  98. A. H. Karapetyan, “Integral Representations for Weighted Spaces of Functions Holomorphic in Tube Domains”, Izv. National Ac. of Sci. of Armenia, Matematika, [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 25(4), 1–19 (1990).

    MathSciNet  Google Scholar 

  99. A. H. Karapetyan, “Integral Representations of Holomorphic Functions in Radial Tube Domains”, Izv. National Ac. of Sci. of Armenia, Matematika, [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 26(6), 1–25 (1991).

    MathSciNet  Google Scholar 

  100. A. H. Karapetyan, “Integral Representations in Tube Domains Over Affine-Homogeneous Cones”, Izv. National Ac. of Sci. of Armenia, Matematika, [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 27(3), 1–21 (1992).

    MATH  MathSciNet  Google Scholar 

  101. A. H. Karapetyan, “Integral Representations in Tube Domains over Polyhedrons and Questions of Approximation of Functions”, AnalysisMath., 20, 185–203 (1994) (Russian).

    MATH  MathSciNet  Google Scholar 

  102. A. H. Karapetyan, “Integral representations of smooth functions in unit ball \( \mathbb{B}^n \) and in ℂn, in: ICM, Abstracts of Short Communications, p. 96. (Zürich, 1994).

  103. A. H. Karapetyan, “Integral representations of smooth functions in ℂn”, in: TFA, Collection of Works Dedicated to the Memory of M. M. Djrbashian, 75–78 (Louys, Yerevan, 1995).

  104. A. H. Karapetyan, “Bounded projections in weighted function spaces in a generalized unit disc”, Ann. Polon. Math., 62(3), 193–218 (1995).

    MATH  MathSciNet  Google Scholar 

  105. A. H. Karapetyan, “Weighted Integral Representations of Holomorphic Functions in Matrix Siegel Domains”, Izv. National Ac. of Sci. of Armenia, Matematika [Journal of ContemporaryMathematical Analysis (National Academy of Sciences of Armenia)], 32(2), 39–56 (1997).

    MathSciNet  Google Scholar 

  106. A. H. Karapetyan, “Weighted \( \bar \partial \) -Integral Representations of C 1-Functions in ℂn”, Dokl. National Akad. Nauk. Armenii, 103(2), 101–104 (2003).

    MathSciNet  Google Scholar 

  107. A. H. Karapetyan, “Weighted \( \bar \partial \) -integral Representations in the Matrix Disc”, Dokl. National Akad. Nauk. of Armenia, 103(3), 199–202 (2003).

    MathSciNet  Google Scholar 

  108. A. H. Karapetyan, “Weighted \( \bar \partial \) -Integral Representations of Smooth Functions in the Siegel Domain”, Izv. National Ac. of Sci. of Armenia, Matematika [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 41(3), 56–70 (2006).

    MathSciNet  Google Scholar 

  109. A. H. Karapetyan, “Weighted \( \bar \partial \) -integral representations in matrix domains”, Complex Variables and Elliptic Equations”, 53(12), 1131–1168 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  110. A. H. Karapetyan, “Weighted Integral Representations of Entire Functions of Several Complex Variables”, Bull. Belg. Math. Soc. Simon Stevin, 15, 289–302 (2008).

    MathSciNet  Google Scholar 

  111. A. V. Harutyunyan, “On Boundedness of a Class of Integral Operators in Anisotropic Spaces of Functions Holomorphic in the Polydisc”, Candidate Thesis Self-Reference (Yerevan, 1992).

  112. A. V. Harutyunyan, “Toeplitz Operators and Division Theorems in Anisotropic Spaces of Holomorphic Functions in the Polydisc”, Complex Variables, 48(4), 347–363 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  113. A. V. Harutyunyan, “Toeplitz Operators and Division Theorems in Spaces H p(ω)”, Complex Variables, 48(10), 803–813 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  114. A. V. Harutyunyan, “Bloch Spaces of Holomorphic Functions in the Polydisc”, J. Funct. Spaces Appl., 5(3), 213–230 (2007).

    MATH  MathSciNet  Google Scholar 

  115. A. V. Harutyunyan, W. Lusky, “On the Boundedness of the Differentiation Operator Between Weighted Spaces of Holomorphic Functions”, Studia Math., 184(3), 233–247 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  116. A. I. Petrosyan, “The Estimate in ℂm-Norm of the Minimal Solutions of \( \bar \partial \) -Equation in Polydisc”, Izv. National Ac. of Sci. of Armenia, Matematika [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 26(2), 99–107 (1991).

    MathSciNet  Google Scholar 

  117. A. I. Petrosyan, “The Weighted Integral Representations of Functions in the Polydisc and in the Space ℂn”, Izv. National Ac. of Sci. of Armenia, Matematika [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 31(1), 38–50 (1996).

    MATH  MathSciNet  Google Scholar 

  118. A. I. Petrosyan, “Derivatives of a Solution of \( \bar \partial \) -Equation in the Bidisc”, Izv. National Ac. of Sci. of Armenia, Matematika [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 38(1), 83–88 (2003).

    MathSciNet  Google Scholar 

  119. A. I. Petrosyan, “Formulas for Derivatives of a Solution of \( \bar \partial \) -Equation in the Ball of ℂn”, in: Topics in Analysis and its Applications, NATO Science Series, II. Mathematics, Physics and Chemistry, Kluwer Academic Publishers, 147, 229–233 (2004).

    MathSciNet  Google Scholar 

  120. A. I. Petrosyan, “On A pω Spaces in the Unit Ball of ℂn”, J. Anal. Appl., 3(1), 47–53 (2005).

    MATH  MathSciNet  Google Scholar 

  121. A. I. Petrosyan, “On Weighted Spaces of Functions Harmonic in ℝn”, Commentationes Mathematicae Universitatis Carolinae, 47(2), 233–240 (2006).

    MATH  MathSciNet  Google Scholar 

  122. A. I. Petrosyan, “On Weighted Classes of Harmonic Functions in the Unit Ball of ℝn”, Complex Var. Theory Appl., 50(12), 953–966 (2005).

    MATH  MathSciNet  Google Scholar 

  123. N. Aronszajn, “La Théorie des Noyaux Reproduistants et ses Applications”, I, Proc. Cambridge Phil. Soc., 39, 133–153 (1943).

    Article  MathSciNet  Google Scholar 

  124. S. Bergman, The Kernel Function and the Conformal Mapping, Math. Surveys, 5 (1950).

  125. N. Aronszajn, “Theory of Reproducing Kernels”, Trans. AMS, 68 337–404 (1950).

    Article  MATH  MathSciNet  Google Scholar 

  126. S. Bergman, The Kernel Function and Conformal Mapping, Second Edition (AMS, Providence, R.I., 1970).

    MATH  Google Scholar 

  127. H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces (Springer-Verlag, Berlin, 2000).

    MATH  Google Scholar 

  128. P. Duren, A. Shuster, Bergman Spaces, Math. Surveys and Monographs, 100 (AMS, Providence, RI., 2004).

    MATH  Google Scholar 

  129. F. Forelly, W. Rudin, “Projections on Spaces of Holomorphic Functions in the Ball”, Indiana Univ. Math. J., 24, 593–602 (1974).

    Article  MathSciNet  Google Scholar 

  130. W. Rudin, Function Theory in the Unit Ball ofn (Springer Verlag, 1980).

  131. F. Ricci, M. Taibleson, “Boundary Values of Harmonic Functions in Mixed Norm Spaces and Their Atomic Structure”, Annali Scuola Normale Superiore — Pisa, Classe di Scienze, Ser. IV, X(1), 1–54 (1983).

    MathSciNet  Google Scholar 

  132. T. G. Genchev, “Paley-Wiener Type Theorems for Functions in Bergman Spaces Over Tube Domains”, J. Math. Anal. Appl., 118, 496–501 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  133. K. L. Avetisyan, “Fractional Integration and Integral Representations in Weighted Classes of Harmonic Functions”, Analysis Math., 26(3), 161–174 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  134. O. Blasco, S. Perez-Esteva, “L p Continuity of Projectors of Weighted Harmonic Bergman Spaces”, Collect. Math., 51(1), 49–58 (2000).

    MATH  MathSciNet  Google Scholar 

  135. P. Charpantier, “Formules Explicition pour les Solutions Minimales de L’equation \( \bar \partial \) u = f dans la Boule et dans le Polydisque de ℂn”, Ann. Inst. Fourier, Grenoble, 30(41), 1921–1954 (1980).

    Google Scholar 

  136. M. Nowak, “Bloch and Möbius Invariant Besov Spaces on the Unit Ball of ℂn”, Complex Variables, 44, 1–12 (2001).

    MATH  Google Scholar 

  137. S. Li, S. Stevic, “Some Characterizations of the Besov Space and the α-Bloch Space”, J. Math. Anal. Appl., 346, 262–273 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  138. S. Li, H. Wulan, “Besov Space in the Unit Ball”, Indian J. Math., 48(2), 177–186 (2006).

    MATH  MathSciNet  Google Scholar 

  139. S. Bochner, “Group Invariance of Cauchy Formula in Several Variables”, Ann. Math., 45, 686–707 (1944).

    Article  MathSciNet  Google Scholar 

  140. I. I. Privalov, Boundary Properties of One-Valued Analytic Functions (MGU, Moscow, 1941). (Russian)

    Google Scholar 

  141. I. I. Privalov, Introduction to the Theory of Functions of Complex Variable (Moscow, 1960). (Russian)

  142. N. I. Akhiezer, The Classical Moment Problem (FM, Moscow, 1961). (Russian)

    Google Scholar 

  143. V. I. Krilov, “On Functions Regular in the Half-Plane”, Mat. sb., 6(48) (1), 95–138 (1939). (Russian)

    Google Scholar 

  144. E. D. Solomentsev, “On Classes of Functions Subharmonic in the Half-Space”, Vestnik MGU, Ser. Mat., 5, 73–91 (1959).

    Google Scholar 

  145. J. A. Shohat, J. D. Tamarkin, The Problem of Moments (AMS, New York, 1943).

    MATH  Google Scholar 

  146. A. L. Shahinyan, “On a Main Inequality in the Function Theory and its Applications”, Izv. Ac. of Sci. of Arm. SSR, Ser. Fiz. Mat., 12(1), 2–25 (1959). (Russian)

    Google Scholar 

  147. R. P. Boas, Entire Functions (Ac. Press, New York, 1954).

    MATH  Google Scholar 

  148. I. S. Kac, “On Integral Representations of Analytic Functions Mapping the Upper Halfplane to its Part”, UMN, 11(3(69)), 139–144 (1956). (Russian)

    MATH  Google Scholar 

  149. Hua Lo-Ken, Harmonic Analysis for Functions of Several Complex Variables in Classical Domains, (IL, Moscow, 1959) (Russian).

    Google Scholar 

  150. H. Shapiro and A. Shields, “On Some Interpolation Problems for Analytic Functions”, Amer. J. Math., 83, 513–532 (1961).

    Article  MATH  MathSciNet  Google Scholar 

  151. S. M. Gindikin, “Analysis in Homogeneous Domains”, Uspekhi Mat. Nauk, 19(4) 3–92 (1964).

    MATH  MathSciNet  Google Scholar 

  152. F. V. Atkinson, Discrete and Continuous Boundary Problems (Ac. Press, Inc., New York, 1964).

    MATH  Google Scholar 

  153. A. A. Goldberg, I. V. Ostrovskii, Value Distribution of Meromorphic Functions (Nauka, Moscow, 1970).

    Google Scholar 

  154. A. L. Shields, D. L. Williams, “Bounded Projections, Duality, and Multipliers in Spaces of Analytic Functions”, Trans. Amer. Math. Soc., 162, 287–302 (1971).

    Article  MathSciNet  Google Scholar 

  155. M. S. Livšic, “Discrete Linear Systems and Their Connections with M. M. Djrbashian Factorization Theory”, Dokl. Akad. Nauk SSSR, 219, 793–796 (1974) (Russian).

    MathSciNet  Google Scholar 

  156. J. H. Shapiro, A. Shields, “Unusual Topological Properties of the Nevanlinna Class”, Amer. J. Math., 97(4), 915–936 (1975).

    Article  MathSciNet  Google Scholar 

  157. M. Tsuji, Potential Theory in Modern Function Theory (Maruzen Co. Ltd., Tokyo, 1975).

    MATH  Google Scholar 

  158. E. Seneta, Regular Varying Functions, Springer Lecture Notes in Mathematics, 508 (Springer, 1976).

  159. M. Stoll, “Mean Value Theorems for Harmonic and Holomorphic Functions on Bounded Symmetric Domains”, J. Reine und Angew. Math., 290, 191–198 (1977).

    MATH  MathSciNet  Google Scholar 

  160. Sh. Dautov, G. M. Henkin, “Zeros of Functions of Finite Order and Weighted Estimates for Solutions of the \( \bar \partial \) -Equation”, Mat. Sb., 107, 371–384 (1978).

    MathSciNet  Google Scholar 

  161. R. R. Coifman, R. Rochberg, “Representation Theorems for Holomorphic and Harmonic Functions in L p”, Asterisque, 77, 11–66 (1980).

    MATH  MathSciNet  Google Scholar 

  162. A. F. Leontyev, Generalizations of Exponential Series (Nauka, Moscow, 1981). (Russian)

    Google Scholar 

  163. B. Yorikke, “AMultidimensionalAnalogy of Privalov’s Theorem”, Math. Nachrichten, 107, 221–233 (1982).

    Article  Google Scholar 

  164. B. Berndtsson, M. Andersson, “Henkin-Ramirez Formulas with Weight Factors”, Ann. Inst. Fourier, 32(3), 91–110 (1982).

    MATH  MathSciNet  Google Scholar 

  165. I. V. Ostrovskii, “On a Hausdorff Moment Problem”, Izv. National Ac. of Sci. of Armenia, Matematika, [Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia)], 25(1), 91–96 (1990).

    MathSciNet  Google Scholar 

  166. S. Saitoh, Integral Transforms, Reproducing Kernels and Their Applications, Pitman Research Notes in Math., 369 (Longman, UK, 1997).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. M. Jerbashian.

Additional information

Original Russian Text © A.M.Jerbashian, V.S.Zakaryan, 2009, published in Izvestiya NAN Armenii. Matematika, 2009, No. 6, pp. 1–24.

About this article

Cite this article

Jerbashian, A.M., Zakaryan, V.S. The contemporary development in M. M. Djrbashian factorization theory and related problems of analysis. J. Contemp. Mathemat. Anal. 44, 341–381 (2009). https://doi.org/10.3103/S1068362309060016

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1068362309060016

MSC2000 numbers

Key words

Navigation