# Positive bases in spaces of polynomials

Positive bases in spaces of polynomials For a nonempty compact set $$\Omega\,\subseteq\,{\mathbb{R}}$$ we determine the maximal possible dimension of a subspace $$X\,\subseteq\,{\mathcal{P}}_{m}(\Omega)$$ of polynomial functions over Ω with degree at most m which possesses a positive basis. The exact value of this maximum depends on topological features of Ω, and we will see that in many of the cases m can be achieved. Whereas only for low m or finite sets Ω it is possible that we have a subspace X with positive basis and with dim X = m + 1. Hence there is no Ω for which a positive basis exists in $${\mathcal{P}}_{m}$$ for all $$m\,\in\,\mathbb{N}$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Positive bases in spaces of polynomials

, Volume 12 (4) – May 1, 2008
19 pages

/lp/springer_journal/positive-bases-in-spaces-of-polynomials-PspHVL8W0d
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-008-2164-4
Publisher site
See Article on Publisher Site

### Abstract

For a nonempty compact set $$\Omega\,\subseteq\,{\mathbb{R}}$$ we determine the maximal possible dimension of a subspace $$X\,\subseteq\,{\mathcal{P}}_{m}(\Omega)$$ of polynomial functions over Ω with degree at most m which possesses a positive basis. The exact value of this maximum depends on topological features of Ω, and we will see that in many of the cases m can be achieved. Whereas only for low m or finite sets Ω it is possible that we have a subspace X with positive basis and with dim X = m + 1. Hence there is no Ω for which a positive basis exists in $${\mathcal{P}}_{m}$$ for all $$m\,\in\,\mathbb{N}$$ .

### Journal

PositivitySpringer Journals

Published: May 1, 2008

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