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Approximation Theory in the Space of Sections of a Vector Bundle
| Content Provider | Scilit |
|---|---|
| Author | Handel, David |
| Copyright Year | 1979 |
| Description | 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. |
| Related Links | https://www.ams.org/tran/1979-256-00/S0002-9947-1979-0546924-X/S0002-9947-1979-0546924-X.pdf |
| Ending Page | 394 |
| Page Count | 12 |
| Starting Page | 383 |
| ISSN | 00029947 |
| e-ISSN | 10886850 |
| DOI | 10.2307/1998117 |
| Journal | Transactions of the American Mathematical Society |
| Volume Number | 256 |
| Language | English |
| Publisher | Duke University Press |
| Publisher Date | 1979-12-01 |
| Access Restriction | Open |
| Subject Keyword | Mathematical Physics Characteristic Classes Vector Bundle Vector Bundles Approximation Theory |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics |
