Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Approximation theory in the space of sections of a vector bundle


Author: David Handel
Journal: Trans. Amer. Math. Soc. 256 (1979), 383-394
MSC: Primary 55R25; Secondary 41A65, 55R40
MathSciNet review: 546924
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Abstract: Let $ p:\,E\, \to \,B$ be a real m-plane bundle and S an n-dimensional subspace of the space of sections $ \Gamma (E)$ of E. S is said to be k-regular if whenever $ {x_1},\, \ldots ,\,{x_k}$ are distinct points of B and $ {\upsilon _i}\, \in \,{p^{ - 1}}({x_i})$, $ 1\, \leqslant \,i\, \leqslant \,k$, there exists a $ \sigma \, \in \,S$ such that $ \sigma ({x_i})\, = \,{\upsilon _i}$ for $ 1\, \leqslant \,i\, \leqslant k$. It is proved that if E has a Riemannian metric and B is compact Hausdorff with at least $ k\, + \,1$ points, then S is k-regular if and only if for each $ \varphi \, \in \,\Gamma (E)$, the set of best approximations to $ \varphi $ by elements of S has dimension at most n - km. This extends a classical theorem of Haar, Kolmogorov, and Rubinstein (the case of the product line bundle). Complex and quaternionic analogues of the above are obtained simultaneously. Existence and nonexistence of k-regular subspaces of a given dimension are obtained in special cases via cohomological methods involving configuration spaces. For example, if E is the product real $ (2m\, - \,1)$-plane bundle over a 2-dimensional disk, then $ \Gamma (E)$ contains a k-regular subspace of dimension $ 2km\, - \,1$, but not one of dimension $ 2km\, - \,1\,\alpha (k)$, where $ \alpha (k)$ denotes the number of ones in the dyadic expansion of k.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0546924-X
Keywords: Vector bundles, best approximation of sections, k-regular subspaces of sections, configuration spaces, characteristic classes
Article copyright: © Copyright 1979 American Mathematical Society