Approximation theory in the space of sections of a vector bundle

Handel, David

doi:10.1090/S0002-9947-1979-0546924-X

Approximation theory in the space of sections of a vector bundle
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Trans. Amer. Math. Soc. 256 (1979), 383-394 Request permission

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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