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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Multipliers and linear functionals for the class $N^{+}$
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by Niro Yanagihara PDF
Trans. Amer. Math. Soc. 180 (1973), 449-461 Request permission

Abstract:

Multipliers for the classes ${H^p}$ are studied recently by several authors, see Duren’s book, Theory of ${H^p}$ spaces, Academic Press, New York, 1970. Here we consider corresponding problems for the class ${N^ + }$ of holomorphic functions in the unit disk such that \[ \lim \limits _{r \to 1} \int _0^{2\pi } {{{\log }^ + }} |f(r{e^{i\theta }})|d\theta = \int _0^{2\pi } {{{\log }^ + }|f({e^{i\theta }})|} d\theta < \infty .\] Our results are: 1. ${N^ + }$ is an $F$-space in the sense of Banach with the distance function \[ \rho (f,g) = \frac {1}{{2\pi }}\int _0^{2\pi } {\log (1 + |f({e^{i\theta }}) - g({e^{i\theta }})|)} d\theta .\] 2. A complex sequence $\Lambda = \{ {\lambda _n}\}$ is a multiplier for ${N^ + }$ into ${H^q}$ for a fixed $q,0 < q < \infty$, if and only if ${\lambda _n} = O(\exp [ - c\sqrt n ])$ for a positive constant $c$. 3. A continuous linear functional $\phi$ on the space ${N^ + }$ is represented by a holomorphic function $g(z) = \Sigma {b_n}{z^n}$ which satisfies ${b_n} = O(\exp [ - c\sqrt n ])$ for a positive constant $c$. Conversely, such a function $g(z) = \Sigma {b_n}{z^n}$ defines a continuous linear functional on the space ${N^ + }$.
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 180 (1973), 449-461
  • MSC: Primary 30A78
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0338382-X
  • MathSciNet review: 0338382