Monomial basis in Korenblum type spaces of analytic functions

Bonet, José; Lusky, Wolfgang; Taskinen, Jari

doi:10.1090/proc/14195

Monomial basis in Korenblum type spaces of analytic functions
HTML articles powered by AMS MathViewer

by José Bonet, Wolfgang Lusky and Jari Taskinen PDF

Proc. Amer. Math. Soc. 146 (2018), 5269-5278 Request permission

Abstract:

It is shown that the monomials $\Lambda =(z^n)_{n=0}^{\infty }$ are a Schauder basis of the Fréchet spaces $A_+^{-\gamma }, \ \gamma \geq 0,$ that consists of all the analytic functions $f$ on the unit disc such that $(1-|z|)^{\mu }|f(z)|$ is bounded for all $\mu > \gamma$. Lusky proved that $\Lambda$ is not a Schauder basis for the closure of the polynomials in weighted Banach spaces of analytic functions of type $H^{\infty }$. A sequence space representation of the Fréchet space $A_+^{-\gamma }$ is presented. The case of (LB)-spaces $A_{-}^{-\gamma }, \ \gamma > 0,$ that are defined as unions of weighted Banach spaces is also studied.

References

Angela A. Albanese, José Bonet, and Werner J. Ricker, The Cesàro operator on Korenblum type spaces of analytic functions, Collect. Math. 69 (2018), no. 2, 263–281. MR 3783155, DOI 10.1007/s13348-017-0205-7
Carlos A. Berenstein and Roger Gay, Complex variables, Graduate Texts in Mathematics, vol. 125, Springer-Verlag, New York, 1991. An introduction. MR 1107514, DOI 10.1007/978-1-4612-3024-3
Klaus D. Bierstedt, José Bonet, and Antonio Galbis, Weighted spaces of holomorphic functions on balanced domains, Michigan Math. J. 40 (1993), no. 2, 271–297. MR 1226832, DOI 10.1307/mmj/1029004753
Klaus D. Bierstedt, José Bonet, and Jari Taskinen, Associated weights and spaces of holomorphic functions, Studia Math. 127 (1998), no. 2, 137–168. MR 1488148, DOI 10.4064/sm-127-2-137-168
Klaus-D. Bierstedt, Reinhold G. Meise, and William H. Summers, Köthe sets and Köthe sequence spaces, Functional analysis, holomorphy and approximation theory (Rio de Janeiro, 1980) North-Holland Math. Stud., vol. 71, North-Holland, Amsterdam-New York, 1982, pp. 27–91. MR 691159
Boris Korenblum, An extension of the Nevanlinna theory, Acta Math. 135 (1975), no. 3-4, 187–219. MR 425124, DOI 10.1007/BF02392019
Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. MR 1758653, DOI 10.1007/978-1-4612-0497-8
Hans Jarchow, Locally convex spaces, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1981. MR 632257, DOI 10.1007/978-3-322-90559-8
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
Wolfgang Lusky, On the Fourier series of unbounded harmonic functions, J. London Math. Soc. (2) 61 (2000), no. 2, 568–580. MR 1760680, DOI 10.1112/S0024610799008443
Reinhold Meise and Dietmar Vogt, Introduction to functional analysis, Oxford Graduate Texts in Mathematics, vol. 2, The Clarendon Press, Oxford University Press, New York, 1997. Translated from the German by M. S. Ramanujan and revised by the authors. MR 1483073
A. L. Shields and D. L. Williams, Bonded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 287–302. MR 283559, DOI 10.1090/S0002-9947-1971-0283559-3
Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. MR 869816
P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991. MR 1144277, DOI 10.1017/CBO9780511608735