DOI: 10.68381/jca20008 ISSN: 0944-6532
Inequalities for Polynomials on the Unit Square via the Krein-Milman Theorem
José Luis Gámez-Merino, Gustavo A. Muñoz-Fernández, Viktor M. Sánchez, Juan B. Seoane-Sepúlveda
We provide sharp Bernstein and Markov inequalities for 2-homogeneous polynomials on the square
\Box\subset {\mathbb R}^2
□
⊂
R
2
with vertices
(0,0)
(
0
,
0
)
,
(1,0)
(
1
,
0
)
,
(1,1)
(
1
,
1
)
and
(0,1)
(
0
,
1
)
. If
{\mathcal P}(^2\Box)
P
(
2
□
)
is the space of such polynomials, we also find the polarization constant of
{\mathcal P}(^2\Box)
P
(
2
□
)
and the unconditional constant for the canonical basis of
{\mathcal P}(^2\Box)
P
(
2
□
)
. All the results are obtained by means of the Krein-Milman Theorem, using a characterization of the extreme 2-homogeneous polynomials on
\Box
□
which is also given in the paper.