Constrained best approximation in Hilbert space. (English) Zbl 0682.41034
Summary: We study the characterization of the solution to the extremal problem
\[
\inf \{\| x\| | x\in C\cap M\},
\]
where x is in a Hilbert space H, C is a convex cone, and M is a translate of a subspace of H determined by interpolation conditions. We introduce a simple geometric property called the “conical hull intersection property” that provides a unifying framework for most of the basic results in the subject of optimal constrained approximation. Our approach naturally lends itself to considering the data cone as opposed to the constraint cone. A nice characterization of the solution occurs, for example, if the data vector associated with M is an interior point of the data cone.
MSC:
| 41A50 | Best approximation, Chebyshev systems |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
| 41A27 | Inverse theorems in approximation theory |
| 41A05 | Interpolation in approximation theory |
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