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Interpolating Blaschke products and angular derivatives
Authors:
Eva A. Gallardo-Gutiérrez and Pamela Gorkin
Journal:
Trans. Amer. Math. Soc. 364 (2012), 2319-2337
MSC (2010):
Primary 46J15, 30J10; Secondary 30H10, 47B38
Posted:
January 3, 2012
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Abstract: We show that to each inner function, there corresponds at least one interpolating Blaschke product whose angular derivatives have precisely the same behavior as the given inner function. We characterize the Blaschke products invertible in the closed algebra  ![$\displaystyle \mbox {has finite angular derivative everywhere}].$](images/img2.gif) We study the most well-known example of a Blaschke product with infinite angular derivative everywhere and show that it is an interpolating Blaschke product. We conclude the paper with a method for constructing thin Blaschke products with infinite angular derivative everywhere.
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- Ahern, P. R.; Clark, D. N., On inner functions with
 -derivative. Michigan Math. J. 21 (1974), 115-127. MR 0344479 (49:9218)
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- Akeroyd, J. R.; Ghatage, P. G.; Tjani, M., Closed-range composition operators on
 and the Bloch space, Integral Equations Operator Theory, 68 (2010), no. 4, 503-517. (electronic) MR 2745476
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- Cargo, G. T., Angular and tangential limits of Blaschke products and their successive derivatives. Canad. J. Math. 14 (1962), 334-348. MR 0136743 (25:204)
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- Chang, Sun Yung A., A characterization of Douglas subalgebras. Acta Math. 137 (1976), no. 2, 82-89. MR 0428044 (55:1074a)
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- Cohn, William S., On the
 classes of derivatives of functions orthogonal to invariant subspaces. Michigan Math. J. 30 (1983), no. 2, 221-229. MR 718268 (84j:30049)
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 containing  . Acta Math. 137 (1976), no. 2, 91-98. MR 0428045 (55:1074b)
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 . Trans. Amer. Math. Soc. 276, 551-581 (1983). MR 688962 (84e:30078)
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Additional Information
Eva A. Gallardo-Gutiérrez
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid Plaza de Ciencias 3 28040, Madrid, Spain
Email:
eva.gallardo@mat.ucm.es
Pamela Gorkin
Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, Pennsylvannia 17837
Email:
pgorkin@bucknell.edu
DOI:
http://dx.doi.org/10.1090/S0002-9947-2012-05535-8
PII:
S 0002-9947(2012)05535-8
Keywords:
Blaschke product,
interpolating Blaschke product,
angular derivative
Received by editor(s):
February 18, 2010
Posted:
January 3, 2012
Additional Notes:
The first author was partially supported by the grant MTM2010-16679 and the Gobierno de Aragón research group Análisis Matemático y Aplicaciones, ref. DGA E-64.
The second author wishes to thank the University of Zaragoza for the support provided by a grant from the research institute IUMA and for its hospitality during the fall of 2009.
Article copyright:
© Copyright 2012 American Mathematical Society
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