Markov's property for kth derivative

Mirosław Baran, Beata Milówka, and Paweł Ozorka. "Markov's property for kth derivative." Annales Polonici Mathematici 106.1 (2012): 31-40. <http://eudml.org/doc/286579>.

@article{MirosławBaran2012,
abstract = {Consider the normed space $(ℙ(ℂ^\{N\}),||·||)$ of all polynomials of N complex variables, where || || a norm is such that the mapping $L_\{g\}: (ℙ(ℂ^\{N\}),||·||) ∋ f ↦ gf ∈ (ℙ(ℂ^\{N\}),||·||)$ is continuous, with g being a fixed polynomial. It is shown that the Markov type inequality $|∂/∂z_\{j\} P|| ≤ M(deg P)^\{m\} ||P||$, j = 1,...,N, $P ∈ ℙ(ℂ^\{N\})$, with positive constants M and m is equivalent to the inequality $||∂^\{N\}/∂z₁...∂z_\{N\} P|| ≤ M^\{\prime \}(deg P)^\{m^\{\prime \}\}||P||$, $P ∈ ℙ(ℂ^\{N\})$, with some positive constants M’ and m’. A similar equivalence result is obtained for derivatives of a fixed order k ≥ 2, which can be more specifically formulated in the language of normed algebras. In addition, we give a nontrivial example of Markov’s inequality in the Wiener algebra of absolutely convergent trigonometric series and show that the Banach algebra approach to Markov’s property furnishes new tools in the study of polynomial inequalities.},
author = {Mirosław Baran, Beata Milówka, Paweł Ozorka},
journal = {Annales Polonici Mathematici},
keywords = {Markov property; polynomial inequalities; normed algebras},
language = {eng},
number = {1},
pages = {31-40},
title = {Markov's property for kth derivative},
url = {http://eudml.org/doc/286579},
volume = {106},
year = {2012},
}

TY - JOUR
AU - Mirosław Baran
AU - Beata Milówka
AU - Paweł Ozorka
TI - Markov's property for kth derivative
JO - Annales Polonici Mathematici
PY - 2012
VL - 106
IS - 1
SP - 31
EP - 40
AB - Consider the normed space $(ℙ(ℂ^{N}),||·||)$ of all polynomials of N complex variables, where || || a norm is such that the mapping $L_{g}: (ℙ(ℂ^{N}),||·||) ∋ f ↦ gf ∈ (ℙ(ℂ^{N}),||·||)$ is continuous, with g being a fixed polynomial. It is shown that the Markov type inequality $|∂/∂z_{j} P|| ≤ M(deg P)^{m} ||P||$, j = 1,...,N, $P ∈ ℙ(ℂ^{N})$, with positive constants M and m is equivalent to the inequality $||∂^{N}/∂z₁...∂z_{N} P|| ≤ M^{\prime }(deg P)^{m^{\prime }}||P||$, $P ∈ ℙ(ℂ^{N})$, with some positive constants M’ and m’. A similar equivalence result is obtained for derivatives of a fixed order k ≥ 2, which can be more specifically formulated in the language of normed algebras. In addition, we give a nontrivial example of Markov’s inequality in the Wiener algebra of absolutely convergent trigonometric series and show that the Banach algebra approach to Markov’s property furnishes new tools in the study of polynomial inequalities.
LA - eng
KW - Markov property; polynomial inequalities; normed algebras
UR - http://eudml.org/doc/286579
ER -