On identities for derivative operators

• Autorzy: Baran Mirosław , Ozorka Paweł

Identyfikatory

DOI

10.5604/01.3001.0013.7230

• Języki publikacji: EN

Abstrakty

EN

Let X be a commutative algebra with unity e and let D be a derivative on X that means the Leibniz rule is satisfied: D(f\cdot g)=D(f)\cdot g+f\cdot D(g). If D^{(k)} is k-th iteration of D then we prove that the following identity holds for any positive integer k frac{1}{k!}\sum\limits_{j=0}^k(-1)^j\binom{k}{j}f^jD^{(m)}(gf^{k-j})=\Phi_{k,m}(g,f)=\begin{cases}0,\ 0\leq m As an application we prove a sharo version of Bernstein’s inequality for trigonometric polynomials.

Słowa kluczowe

EN

derivative operator; polynomial inequality

PL

operator instrumentów pochodnych; nierówność wielomianowa

• Wydawca: Państwowa Wyższa Szkoła Zawodowa w Tarnowie
• Czasopismo: Science, Technology and Innovation
• Rocznik: 2019
• Tom: Vol. 7, no. 4
• Strony: 13--16
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Baran Mirosław

• Faculty of Mathematics, Physics and Technical Science, Pedagogical University, Podchorążych 2, 30-084 Kraków, Poland

autor

Ozorka Paweł

• Department of Mathematics, Faculty of Mathematical and Natural Sciences, University of Applied Sciences in Tarnow, Mickiewicza 8, 33-100 Tarnów, Poland

Bibliografia

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• [5] M. Baran, A. Kowalska, B. Milówka, P. Ozorka, Identities for a derivation operator and their applications, Dolomites Res. Notes Approx. 8 (2015), Special Issue, 102-110.
• [6] M. Baran, B. Milówka, P. Ozorka, Markov’s property for k-th derivative, Annales Polonici Mathematici, 106 (2012), 31–40.
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• [11] W. Rudin, Functional Analysis, McGraw-Hill Book
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