Approximation theory in the space of sections of a vector bundle
Author:
David Handel
Journal:
Trans. Amer. Math. Soc. 256 (1979), 383394
MSC:
Primary 55R25; Secondary 41A65, 55R40
MathSciNet review:
546924
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Abstract: Let be a real mplane bundle and S an ndimensional subspace of the space of sections of E. S is said to be kregular if whenever are distinct points of B and , , there exists a such that for . It is proved that if E has a Riemannian metric and B is compact Hausdorff with at least points, then S is kregular if and only if for each , the set of best approximations to 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 kregular subspaces of a given dimension are obtained in special cases via cohomological methods involving configuration spaces. For example, if E is the product real plane bundle over a 2dimensional disk, then contains a kregular subspace of dimension , but not one of dimension , where denotes the number of ones in the dyadic expansion of k.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719790546924X
PII:
S 00029947(1979)0546924X
Keywords:
Vector bundles,
best approximation of sections,
kregular subspaces of sections,
configuration spaces,
characteristic classes
Article copyright:
© Copyright 1979
American Mathematical Society
