### Abstract

The C^{1}-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 H_{1} ⊂ H_{2} ⊂ ⋯ ⊂ ℍ such that ∪_{n≥1} H_{n} is dense in span{X} and π_{n} (X) = X ∩ H_{n} for each n ≥ 1. Here, π_{n} : ℍ → H_{n} is the orthogonal projection. It is also shown that when X is compact convex with span{X} = ℍ and satisfies the above condition, then C^{1} (X) is complete if and only if the C^{1}-Whitney extension theorem holds for X. Finally, for compact subsets of ℍ, an extension of the C^{1}-Weierstrass approximation theorem is proved for C^{1} maps ℍ → ℍ with compact derivatives.

Original language | English (US) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

^{1}-Weierstrass for compact sets in Hilbert space.

*Journal of Mathematical Analysis and Applications*,

*285*(1), 299-320. https://doi.org/10.1016/S0022-247X(03)00427-X

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^{1}-Weierstrass for compact sets in Hilbert space',

*Journal of Mathematical Analysis and Applications*, vol. 285, no. 1, pp. 299-320. https://doi.org/10.1016/S0022-247X(03)00427-X

**C ^{1}-Weierstrass for compact sets in Hilbert space.** / Movahedi-Lankarani, Hossein; Wells, R.

Research output: Contribution to journal › Article

TY - JOUR

T1 - C1-Weierstrass for compact sets in Hilbert space

AU - Movahedi-Lankarani, Hossein

AU - Wells, R.

PY - 2003/9/1

Y1 - 2003/9/1

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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0141515684&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0141515684&partnerID=8YFLogxK

U2 - 10.1016/S0022-247X(03)00427-X

DO - 10.1016/S0022-247X(03)00427-X

M3 - Article

VL - 285

SP - 299

EP - 320

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -

^{1}-Weierstrass for compact sets in Hilbert space. Journal of Mathematical Analysis and Applications. 2003 Sep 1;285(1):299-320. https://doi.org/10.1016/S0022-247X(03)00427-X