C1-Weierstrass for compact sets in Hilbert space

Research output: Contribution to journalArticle

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Abstract

The C1-Weierstrass approximation theorem is proved for any compact subset X of a Hilbert space ℍ. The same theorem is also proved for Whitney 1-jets on X when X satisfies the following further condition: There exist finite dimensional linear subspaces H1 ⊂ H2 ⊂ ⋯ ⊂ ℍ such that ∪n≥1 Hn is dense in span{X} and πn (X) = X ∩ Hn for each n ≥ 1. Here, πn : ℍ → Hn is the orthogonal projection. It is also shown that when X is compact convex with span{X} = ℍ and satisfies the above condition, then C1 (X) is complete if and only if the C1-Whitney extension theorem holds for X. Finally, for compact subsets of ℍ, an extension of the C1-Weierstrass approximation theorem is proved for C1 maps ℍ → ℍ with compact derivatives.

Original languageEnglish (US)
Pages (from-to)299-320
Number of pages22
JournalJournal of Mathematical Analysis and Applications
Volume285
Issue number1
DOIs
StatePublished - Sep 1 2003

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Hilbert spaces
Set theory
Compact Set
Hilbert space
Weierstrass Theorem
Derivatives
Approximation Theorem
Subset
Extension Theorem
Orthogonal Projection
Subspace
If and only if
Derivative
Theorem

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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abstract = "The C1-Weierstrass approximation theorem is proved for any compact subset X of a Hilbert space ℍ. The same theorem is also proved for Whitney 1-jets on X when X satisfies the following further condition: There exist finite dimensional linear subspaces H1 ⊂ H2 ⊂ ⋯ ⊂ ℍ such that ∪n≥1 Hn is dense in span{X} and πn (X) = X ∩ Hn for each n ≥ 1. Here, πn : ℍ → Hn is the orthogonal projection. It is also shown that when X is compact convex with span{X} = ℍ and satisfies the above condition, then C1 (X) is complete if and only if the C1-Whitney extension theorem holds for X. Finally, for compact subsets of ℍ, an extension of the C1-Weierstrass approximation theorem is proved for C1 maps ℍ → ℍ with compact derivatives.",
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C1-Weierstrass for compact sets in Hilbert space. / Movahedi-Lankarani, Hossein; Wells, R.

In: Journal of Mathematical Analysis and Applications, Vol. 285, No. 1, 01.09.2003, p. 299-320.

Research output: Contribution to journalArticle

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N2 - The C1-Weierstrass approximation theorem is proved for any compact subset X of a Hilbert space ℍ. The same theorem is also proved for Whitney 1-jets on X when X satisfies the following further condition: There exist finite dimensional linear subspaces H1 ⊂ H2 ⊂ ⋯ ⊂ ℍ such that ∪n≥1 Hn is dense in span{X} and πn (X) = X ∩ Hn for each n ≥ 1. Here, πn : ℍ → Hn is the orthogonal projection. It is also shown that when X is compact convex with span{X} = ℍ and satisfies the above condition, then C1 (X) is complete if and only if the C1-Whitney extension theorem holds for X. Finally, for compact subsets of ℍ, an extension of the C1-Weierstrass approximation theorem is proved for C1 maps ℍ → ℍ with compact derivatives.

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