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Universal Blaschke products

Published online by Cambridge University Press:  15 January 2004

PAMELA GORKIN
Affiliation:
Department of Mathematics, Bucknell University, PA 17837, Lewisburg, U.S.A. e-mail: pgorkin@bucknell.edu
RAYMOND MORTINI
Affiliation:
Département de Mathématiques, Université de Metz, Ile du Saulcy, F-57045 Metz, France. e-mail: mortini@poncelet.univ-metz.fr

Abstract

We extend a result of M. Heins by showing that for any sequence of points $(z_n)$ in the unit disk ${\Bbb D}$ tending to the boundary, there is a Blaschke product $B$ which is universal for noneuclidian translates in the sense that the set $\{B((z\,{+}\,z_n)/(1\,{+}\,\overline{z}_nz))\,{:} n\,{\in}\,{\Bbb N}\}$ is locally uniformly dense in the set of all holomorphic functions bounded by one on ${\Bbb D}$. From this, we conclude that for every countable set ${\sp L}$ of hyperbolic/parabolic automorphisms of the unit disk there exists a Blaschke product which is a common cyclic vector in $H^2$ for the composition operators associated with the elements in ${\sp L}$. These results are obtained by transferring the associated approximation problems to interpolation problems on the corona of $H^\infty$.

Type
Research Article
Copyright
2004 Cambridge Philosophical Society

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