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 language | English (US) |
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Pages (from-to) | 299-320 |
Number of pages | 22 |
Journal | Journal of Mathematical Analysis and Applications |
Volume | 285 |
Issue number | 1 |
DOIs | |
State | Published - Sep 1 2003 |
All Science Journal Classification (ASJC) codes
- Analysis
- Applied Mathematics