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Transactions of the American Mathematical Society

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

 
 

 

Some $H^{\infty }$-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
DOI: https://doi.org/10.1090/S0002-9947-1975-0372219-X
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: <IMG WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$K$">-curve, <IMG WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$K$">-function, Wermer map, part, <!– MATH ${H^\infty }$ –> <IMG WIDTH="41" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${H^\infty }$">, interpolating sequence, Blaschke product
Article copyright: © Copyright 1975 American Mathematical Society