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Transactions of the American Mathematical Society

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



Some $ H\sp{\infty }$-interpolating sequences and the behavior of certain of their Blaschke products

Author: Max L. Weiss
Journal: Trans. Amer. Math. Soc. 209 (1975), 211-223
MSC: Primary 30A98; Secondary 46J15
MathSciNet review: 0372219
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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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Keywords: $ K$-curve, $ K$-function, Wermer map, part, $ {H^\infty }$, interpolating sequence, Blaschke product
Article copyright: © Copyright 1975 American Mathematical Society