C1-Weierstrass for compact sets in Hilbert space

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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

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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