Extending piecewise polynomial functions in two variables

Andreas Fischer[1]; Murray Marshall[2]

  • [1] Fields Institute, Toronto, Canada, current: Comenius Gymnasium Datteln, Südring150, 45711 Datteln, Germany
  • [2] University of Saskatchewan, Department of Mathematics & Statistics, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada

Annales de la faculté des sciences de Toulouse Mathématiques (2013)

  • Volume: 22, Issue: 2, page 253-268
  • ISSN: 0240-2963

Abstract

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We study the extensibility of piecewise polynomial functions defined on closed subsets of ℝ 2 to all of ℝ 2 . The compact subsets of ℝ 2 on which every piecewise polynomial function is extensible to ℝ 2 can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of ℝ . Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise polynomial function defined on a compact, convex, but undefinable subset of ℝ 2 which is not extensible to ℝ 2 .

How to cite

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Fischer, Andreas, and Marshall, Murray. "Extending piecewise polynomial functions in two variables." Annales de la faculté des sciences de Toulouse Mathématiques 22.2 (2013): 253-268. <http://eudml.org/doc/275299>.

@article{Fischer2013,
abstract = {We study the extensibility of piecewise polynomial functions defined on closed subsets of $\mathbb\{R\}^2$ to all of $\mathbb\{R\}^2$. The compact subsets of $\mathbb\{R\}^2$ on which every piecewise polynomial function is extensible to $\mathbb\{R\}^2$ can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of $\mathbb\{R\}$. Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise polynomial function defined on a compact, convex, but undefinable subset of $\mathbb\{R\}^2$ which is not extensible to $\mathbb\{R\}^2$.},
affiliation = {Fields Institute, Toronto, Canada, current: Comenius Gymnasium Datteln, Südring150, 45711 Datteln, Germany; University of Saskatchewan, Department of Mathematics & Statistics, 106 Wiggins Road, Saskatoon, SK, S7N 5E6, Canada},
author = {Fischer, Andreas, Marshall, Murray},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {piecewise polynomial function; semi-algebraic set; -minimal expansion; definable set; Lipschitz condition; local convexity},
language = {eng},
month = {6},
number = {2},
pages = {253-268},
publisher = {Université Paul Sabatier, Toulouse},
title = {Extending piecewise polynomial functions in two variables},
url = {http://eudml.org/doc/275299},
volume = {22},
year = {2013},
}

TY - JOUR
AU - Fischer, Andreas
AU - Marshall, Murray
TI - Extending piecewise polynomial functions in two variables
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2013/6//
PB - Université Paul Sabatier, Toulouse
VL - 22
IS - 2
SP - 253
EP - 268
AB - We study the extensibility of piecewise polynomial functions defined on closed subsets of $\mathbb{R}^2$ to all of $\mathbb{R}^2$. The compact subsets of $\mathbb{R}^2$ on which every piecewise polynomial function is extensible to $\mathbb{R}^2$ can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of $\mathbb{R}$. Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise polynomial function defined on a compact, convex, but undefinable subset of $\mathbb{R}^2$ which is not extensible to $\mathbb{R}^2$.
LA - eng
KW - piecewise polynomial function; semi-algebraic set; -minimal expansion; definable set; Lipschitz condition; local convexity
UR - http://eudml.org/doc/275299
ER -

References

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