Computational Methods and Function Theory 3 (2003), No. 1, 151--164
Copyright Heldermann Verlag 2003
Maude Giasson
maude.giasson@math.com, Université de Laval, Département de Mathématiques et Statistique, Québec G1K 7P4, Canada.
Walter Hengartner†
Université de Laval, Département de Mathématiques et Statistique, Québec G1K 7P4, Canada.
Gerhard Opfer
opfer@math.uni-hamburg.de, Universität Hamburg, Fachbereich Mathematik, Bundesstraße 55, 20146 Hamburg, Germany.
Some of the known Haar spaces are linear hulls of shifts of a single function $G$ on $\mathbb{C} \backslash \{0\}$. We study $N$-dimensional and universal analytic Haar space generators for some closed sets $F$ of $\mathbb{C}$ (in the sense that an arbitrary finite number of shifts generates Haar spaces by forming linear hulls). The suitable function space for our investigation is $\mathrm{C}^{\circ}(F)$, the space of all complex valued, continuous functions $f$ on $F$ with the defining property $\lim_{z\in F,z\to\infty}f(z)=0$. In many cases we are able to characterize universal Haar space generators. We show, in addition, that in $\mathrm{C}^{\circ}(F)$ a best approximation by elements of finite dimensional spaces $V$ is unique if and only if $V$ is a Haar space.
Keywords: Complex Haar spaces, shift generated spaces, approximation on unbounded domains, Haar space generators.
MSC 2000: 30C15, 30E10, 41A45, 41A50, 41A52.
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