Markov's property for $k$th derivative

Tom 106 / 2012

Mirosław Baran, Beata Milówka, Paweł Ozorka Annales Polonici Mathematici 106 (2012), 31-40 MSC: Primary 31C10; Secondary 32U35, 41A17. DOI: 10.4064/ap106-0-3

Streszczenie

Consider the normed space $(\mathbb P(\mathbb C^N),\|\cdot \| )$ of all polynomials of $N$ complex variables, where $\|\,\|$ a norm is such that the mapping $L_g:(\mathbb P(\mathbb C^N),\|\cdot \| )\ni f\mapsto gf\in(\mathbb P(\mathbb C^N),\|\cdot \| )$ is continuous, with $g$ being a fixed polynomial. It is shown that the Markov type inequality $$ \left\| \frac{\partial}{\partial z_j}P\right\|\leq M(\deg P)^m\| P\|,\quad\ j=1, \dots, N, \, P\in \mathbb P(\mathbb C^N), $$ with positive constants $M$ and $m$ is equivalent to the inequality $$ \left\| \frac{\partial^N}{\partial z_1\dots \partial z_N}P\right\|\leq M'(\deg P)^{m'}\| P\|,\quad\ P\in \mathbb P(\mathbb C^N) , $$ with some positive constants $M'$ and $m'$. A similar equivalence result is obtained for derivatives of a fixed order $k\geq 2$, 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.

Autorzy

  • Mirosław BaranInstitute of Mathematics
    Jagiellonian University
    Łojasiewicza 6
    30-348 Kraków, Poland
    e-mail
  • Beata MilówkaInstitute of Mathematical and Natural Sciences
    State Higher Vocational School in Tarnów
    Mickiewicza 8
    33-100 Tarnów, Poland
    e-mail
  • Paweł OzorkaInstitute of Mathematical and Natural Sciences
    State Higher Vocational School in Tarnów
    Mickiewicza 8
    33-100 Tarnów, Poland
    e-mail

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