Markov's property for kth derivative

Mirosław Baran ; Beata Milówka ; Paweł Ozorka

Annales Polonici Mathematici, Tome 105 (2012), p. 31-40 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

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Publié le : 2012-01-01

Zbl 1254.31008

EUDML-ID : urn:eudml:doc:286579