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Shift Generated Haar Spaces on Compact Domains in the Complex Plane.
- Source :
-
Constructive Approximation . 2005, Vol. 22 Issue 1, p113-132. 20p. - Publication Year :
- 2005
-
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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 01764276
- Volume :
- 22
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Constructive Approximation
- Publication Type :
- Academic Journal
- Accession number :
- 16731318
- Full Text :
- https://doi.org/10.1007/s00365-004-0570-9