Markov's property for kth derivative

• Autorzy: Mirosław Baran , Beata Milówka , Paweł Ozorka
• Języki publikacji: EN

Abstrakty

EN

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
$||∂^{N}/∂z₁...∂z_{N} P|| ≤ M'(deg P)^{m'}||P||$, $P ∈ ℙ(ℂ^{N})$,
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.

Kategorie tematyczne

• 32U35: Pluricomplex Green functions
• 31C10: Pluriharmonic and plurisubharmonic functions
• 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol\cprime ski{\u\i}-type inequalities)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2012
• Tom: 106
• Numer: 1
• Strony: 31-40

Daty

wydano

2012

Twórcy

autor

Mirosław Baran

• Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

autor

Beata Milówka

• Institute of Mathematical and Natural Sciences, State Higher Vocational School in Tarnów, Mickiewicza 8, 33-100 Tarnów, Poland

autor

Paweł Ozorka

• Institute of Mathematical and Natural Sciences, State Higher Vocational School in Tarnów, Mickiewicza 8, 33-100 Tarnów, Poland

Identyfikatory

DOI

10.4064/ap106-0-3