Some -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 be a strictly increasing continuous real function defined near with . Such a function is called a -function if for every constant as . The curve in the open unit disc with corresponding representation is called a -curve. Several analytic and geometric conditions are obtained for -curves and -functions. This provides a framework for some rather explicit results involving parts in the closure of -curves, -interpolating sequences lying on -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.

**[1]**Kenneth Hoffman,*Banach spaces of analytic functions*, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR**0133008****[2]**Kenneth Hoffman,*Bounded analytic functions and Gleason parts*, Ann. of Math. (2)**86**(1967), 74–111. MR**0215102****[3]**U. V. Satyanarayana and Max L. Weiss,*The geometry of convex curves tending to 1 in the unit disc*, Proc. Amer. Math. Soc.**41**(1973), 159–166. MR**0318497**, 10.1090/S0002-9939-1973-0318497-8**[4]**Dennis H. Wortman,*Interpolating sequences on convex curves in the open unit disc*, Proc. Amer. Math. Soc.**48**(1975), 157–164. MR**0361092**, 10.1090/S0002-9939-1975-0361092-7

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DOI:
https://doi.org/10.1090/S0002-9947-1975-0372219-X

Keywords:
-curve,
-function,
Wermer map,
part,
,
interpolating sequence,
Blaschke product

Article copyright:
© Copyright 1975
American Mathematical Society