Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1975-0361575-X
MathSciNet review: 0361575
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] V. I. Averbuh and O. G. Smoljanov, Differentiation theory in linear topological spaces, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 201-260 = Russian Math. Surveys 22 ( 1967), no. 6, 201-258. MR 36 #6933. MR 0223886 (36:6933)
  • [2] J. Dieudonné, Foundations of modern analysis, Pure and Appl. Math., vol. 10, Academic Press, New York, 1960. MR 22 #11074. MR 0120319 (22:11074)
  • [3] J. R. Rice, The approximation of functions. Vol. II. Nonlinear and multivariate theory, Addison-Wesley, Reading, Mass., 1969. MR 39 #5989. MR 0244675 (39:5989)
  • [4] E. R. Rozema and P. W. Smith, Nonlinear approximation in uniformly smooth Banach spaces, Trans. Amer. Math. Soc. 188 (1974), 199-212. MR 0330875 (48:9212)
  • [5] I. Singer, Best approximation in normed vector spaces by elements of vector subspaces, Ed. Acad. Repub. Soc. Romania, Bucharest, 1967; English Transl., Die Grundlehren der math. Wissenschaften, Band 171, Springer-Verlag, New York and Berlin, 1970. MR 38 #3677; 42 #4937. MR 0235368 (38:3677)
  • [6] D. E. Wulbert, Uniqueness and differential characterization of approximations from manifolds of functions, Amer. J. Math. 93 (1971), 350-366. MR 45 #4036. MR 0294968 (45:4036)
  • [7] -, Nonlinear approximation with tangential characterization, Amer. J. Math. 93 (1971), 718-730. MR 45 #4037. MR 0294969 (45:4037)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 41A65

Retrieve articles in all journals with MSC: 41A65


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0361575-X
Keywords: Nonlinear approximation, curvature, unique best approximation, splines, Hilbert space
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society