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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Unique best nonlinear approximation in Hilbert spaces

Authors: Charles K. Chui and Philip W. Smith
Journal: Proc. Amer. Math. Soc. 49 (1975), 66-70
MSC: Primary 41A65
MathSciNet review: 0361575
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Abstract: Using the notion of curvature of a manifold, developed by J. R. Rice and recently studied by E. R. Rozema and the second named author, the authors prove the following result: Let $ H$ be a Hilbert space and $ F$ map $ {R^n}$ into $ H$ such that $ F$ is a homeomorphism onto $ \mathfrak{F} = F({R^n})$ and is twice continuously Fréchet differentiable. Then if $ F'(\alpha ) \cdot {R^n}$ is of dimension $ n$ for all $ \alpha \in {R^n}$, the manifold $ \mathfrak{F}$ has finite curvature everywhere. It follows that there is a neighborhood $ \mathfrak{U}$ of $ \mathfrak{F}$ such that each $ u \in \mathfrak{U}$ has a unique best approximation from $ \mathfrak{F}$. However, these results do not hold in general for uniformly smooth Banach spaces.

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Keywords: Nonlinear approximation, curvature, unique best approximation, splines, Hilbert space
Article copyright: © Copyright 1975 American Mathematical Society