Positivity 12 (2008), 691–709
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040691-19, published online May 1, 2008
Positive bases in spaces of polynomials
B´alint Farkas and Szil´ard Gy. R´ev´esz
Abstract. For a nonempty compact set Ω ⊆ R we determine the maximal
possible dimension of a subspace X ⊆P
(Ω) of polynomial functions over
Ωwithdegreeatmostm 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 P
for all m ∈ N.
Mathematics Subject Classiﬁcation (2000). Primary 41A17; Secondary 30E10,
Keywords. Positive polynomials, positive bases, Bernstein-Lorentz represen-
tation, Bernstein-Lorentz degree.
Consider the interval I =[−1, 1] and the space P
of algebraic polynomials with
degree at most n ∈ N over I. In many problems, e.g., in approximation theory, it
is desirable to represent a given polynomial p ∈
=: P in terms of positive
linear combinations of preliminary given, positive polynomials (positive means here
and in the following that the polynomial is pointwise nonnegative). For example
consider the Bernstein polynomials
(1 + x)
for k =0,...,n,
which are evidently positive on [−1, 1]. The following is classical.
Proposition. The system E
} is a basis of P
named author was supported in part by the Hungarian National Foundation for Scien-
tiﬁc Research, Project #s T-049301, T-049693 and K-61908.
This work was accomplished during the 2
author’s stay in Paris under his Marie Curie fellow-
ship, contract # MEIF-CT-2005-022927.