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2 2019

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On Vladimir Markov type inequality in Lp norms on the interval [-1; 1]

Baran Mirosław, Ozorka Paweł

Science, Technology and Innovation

2019

Vol. 7, no. 4

9--12

We prove inequality ||P(k)||Lp(-1;1)≤Bp||Tn(k)||Lp(-1;1)n^(2/p) ||P||Lp(-1;1); where Bp are constants independent of n = deg P with 1 ≤ p ≤ 2, which is sharp in the case k ≥ 3. A method presented in this note is based on a factorization of linear operator of k-th derivative throughout normed spaces of polynomial equipped with a Wiener type norm.

On identities for derivative operators

Baran Mirosław, Ozorka Paweł

Science, Technology and Innovation

2019

Vol. 7, no. 4

13--16

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.