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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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Some $H^{\infty }$-interpolating sequences and the behavior of certain of their Blaschke products
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by Max L. Weiss PDF
Trans. Amer. Math. Soc. 209 (1975), 211-223 Request permission

Abstract:

Let $f$ be a strictly increasing continuous real function defined near ${0^ + }$ with $f(0) = f’(0) = 0$. Such a function is called a $K$-function if for every constant $k,f(\theta + kf(\theta ))/f(\theta ) \to 1/$ as $\theta \to {0^ + }$. The curve in the open unit disc with corresponding representation $1 - r = f(\theta )$ is called a $K$-curve. Several analytic and geometric conditions are obtained for $K$-curves and $K$-functions. This provides a framework for some rather explicit results involving parts in the closure of $K$-curves, ${H^\infty }$-interpolating sequences lying on $K$-curves and the behavior of their Blaschke products. In addition, a sequence of points in the disc tending upper tangentially to 1 with moduli increasing strictly to 1 and arguments decreasing strictly to 0 is proved to be interpolating if and only if the hyperbolic distance between successive points remains bounded away from zero.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 209 (1975), 211-223
  • MSC: Primary 30A98; Secondary 46J15
  • DOI: https://doi.org/10.1090/S0002-9947-1975-0372219-X
  • MathSciNet review: 0372219