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 be a real *m*-plane bundle and *S* an *n*-dimensional subspace of the space of sections of *E*. *S* is said to be *k*-regular 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 *k*-regular 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 *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 -plane bundle over a 2-dimensional disk, then contains a *k*-regular subspace of dimension , but not one of dimension , where 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