Shift Generated Haar Spaces on Compact Domains in the Complex Plane |
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Authors: | Walter Hengartner Gerhard Opfer |
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Institution: | (1) Université de Laval, Département de Mathématiques, Québec G1K 7P4, Canada |
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Abstract: | Haar spaces are certain finite-dimensional subspaces of $\cc(K)$, where $K$
is a compact set and $\cc(K)$ is the Banach space of continuous functions
defined on $K$ having values in $\C$. We characterize those Haar spaces
which are generated by shifts applied to a single, analytic
function for $K\subset\C$. This means that an arbitrary finite
number of shifts generates Haar spaces by forming linear hulls.
We have to distinguish two cases: (a) $K\not=\overline{K^\circ}$;
(b) $K=\overline{K^\circ}$. It turns out that, in case (a),
an analytic Haar space generator for dimensions one and two
is already a universal Haar space generator for all dimensions.
The geometrically simplest case that, in case (b), $K$ is convex
with smooth boundary turns out to be the most difficult case.
There is one numerical example in which the entire function $f:=1/\Gamma$
is interpolated in a shift generated Haar space of dimension four. |
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Keywords: | Complex Haar spaces Complex approximation Shift generated Haar spaces |
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