Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Mean-square approximation by polynomials on the unit disk

Authors: Thomas L. Kriete and Barbara D. MacCluer
Journal: Trans. Amer. Math. Soc. 322 (1990), 1-34
MSC: Primary 30E10; Secondary 30D50, 46E20
MathSciNet review: 948193
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate $ {P^2}(\mu )$, the closure in $ {L^2}(\mu )$ of the complex polynomials, for certain measures $ \mu $ on the closed unit disk in the complex plane. Specifically, we present a condition on $ \mu $ which guarantees that $ {P^2}(\mu )$ decomposes into a natural direct sum.

References [Enhancements On Off] (What's this?)

  • [B1] James E. Brennan, Point evaluations, invariant subspaces and approximation in the mean by polynomials, J. Funct. Anal. 34 (1979), no. 3, 407–420. MR 556263, 10.1016/0022-1236(79)90084-3
  • [B2] James E. Brennan, Weighted polynomial approximation, quasianalyticity and analytic continuation, J. Reine Angew. Math. 357 (1985), 23–50. MR 783532, 10.1515/crll.1985.357.23
  • [Ca] T. Carleman, Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen, Ark. Mat. Astronom. Fys. 17 (1923), 1-30.
  • [Cl] S. Clary, Quasi-similarity and subnormal operators, Doctoral Dissertation, Univ. of Michigan, 1973.
  • [Ha] William W. Hastings, A construction of Hilbert spaces of analytic functions, Proc. Amer. Math. Soc. 74 (1979), no. 2, 295–298. MR 524303, 10.1090/S0002-9939-1979-0524303-4
  • [H] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR 0133008
  • [Hr1] S. V. Hruščëv, Removal of singularities for Cauchy integrals and an analogue of the Hinčin-Ostrovskiĭ theorem for sequences of increasing functions, Dokl. Akad. Nauk SSSR 214 (1974), 524–527 (Russian). MR 0350012
  • [Hr2] S. V. Hruščëv, The problem of simultaneous approximation and of removal of the singularities of Cauchy type integrals, Trudy Mat. Inst. Steklov. 130 (1978), 124–195, 223 (Russian). Spectral theory of functions and operators. MR 505685
  • [Hr3] S. V. Hruščev, The Brennan alternative for measures with finite entropy, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 14 (1979), no. 3, 184–191, 234 (Russian, with English summary). MR 553495
  • [Ke] È. M. Kegejan, Simultaneous approximation in a disc, Izv. Akad. Nauk Armjan. SSR Ser. Mat. 1 (1966), no. 5, 317–330 (Russian, with Armenian and English summaries). MR 0209487
  • [Kd] M. Keldych, Sur l’approximation en moyenne par polynomes des fonctions d’une variable complexe, Rec. Math. [Mat. Sbornik] N. S. 16(58) (1945), 1–20 (French., with Russian summary). MR 0012717
  • [Kr1] Thomas Kriete, On the structure of certain 𝐻²(𝜇) spaces, Indiana Univ. Math. J. 28 (1979), no. 5, 757–773. MR 542335, 10.1512/iumj.1979.28.28053
  • [Kr2] -, Splitting and boundary behavior in certain $ {H^2}(\mu )$ spaces, in Linear and Complex Analysis Problem Book (V. Havin et al., eds.), Lecture Notes in Math., vol. 1043, Springer-Verlag, New York, 1984.
  • [MS] Barbara D. MacCluer and Joel H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), no. 4, 878–906. MR 854144, 10.4153/CJM-1986-043-4
  • [Ne] Rolf Nevanlinna, Analytic functions, Translated from the second German edition by Phillip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. MR 0279280
  • [N] N. K. \cyr{N}ikol′skiĭ, Izbrannye zadachi vesovoi approksimatsii i spektralnogo analiza, Izdat. “Nauka” Leningrad. Otdel., Leningrad, 1974 (Russian). Trudy Mat. Inst. Steklov. 120 (1974). MR 0467269
  • [RR] Marvin Rosenblum and James Rovnyak, Change of variables formulas with Cayley inner functions, Topics in functional analysis (essays dedicated to M. G. Kreĭn on the occasion of his 70th birthday), Adv. in Math. Suppl. Stud., vol. 3, Academic Press, New York-London, 1978, pp. 283–320. MR 538025
  • [Ry] John V. Ryff, Subordinate 𝐻^{𝑝} functions, Duke Math. J. 33 (1966), 347–354. MR 0192062
  • [Sc] H. Schwartz, Composition operators on $ {H^p}$, Doctoral Dissertation, University of Toledo, Ohio, 1969.
  • [Sh] Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375–404. MR 881273, 10.2307/1971314
  • [Tr1] Tavan T. Trent, 𝐻²(𝜇) spaces and bounded point evaluations, Pacific J. Math. 80 (1979), no. 1, 279–292. MR 534718
  • [Tr2] Tavan T. Trent, A characterization of 𝑃²(𝜇)̸=𝐿²(𝜇), J. Funct. Anal. 64 (1985), no. 2, 163–177. MR 812389, 10.1016/0022-1236(85)90072-2
  • [V1] A. L. Vol′berg, Mean quadratic completeness of polynomials beyond the limits of Szegő’s theorem, Dokl. Akad. Nauk SSSR 241 (1978), no. 3, 521–524 (Russian). MR 504220
  • [V2] A. L. Vol′berg, Simultaneous approximation by polynomials on the circle and in the interior of the disc, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 92 (1979), 60–84, 319 (Russian, with English summary). Investigations on linear operators and the theory of functions, IX. MR 566742
  • [V3] A. L. Vol′berg, Summability of the logarithm of a quasi-analytic function, Dokl. Akad. Nauk SSSR 265 (1982), no. 6, 1297–1302 (Russian). MR 670692
  • [VE] A. L. Vol′berg and B. Ërikke, Summability of the logarithm of an almost analytic function and generalization of the Levinson-Cartwright theorem, Mat. Sb. (N.S.) 130(172) (1986), no. 3, 335–348, 431 (Russian). MR 865765
  • [W] J. Wermer, On a class of normed rings, Ark. Mat. 2 (1954), 537–551. MR 0062363

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30E10, 30D50, 46E20

Retrieve articles in all journals with MSC: 30E10, 30D50, 46E20

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society