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Izv. RAN. Ser. Mat., 1998, Volume 62, Issue 6, Pages 125–142 (Mi izv223)  

This article is cited in 2 scientific papers (total in 2 papers)

Fractional derivatives and inequalities for trigonometric polynomials in spaces with asymmetric norms

A. I. Kozko

M. V. Lomonosov Moscow State University

Abstract: We consider the Bernstein–Jackson–Nikol'skii inequalities for fractional derivatives in the case when the norm is asymmetric. Assume that $n\in\mathbb N$, $p_1,p_2,q_1,q_2\in[1,\infty]$, and $\alpha\in\mathbb R_+$. Then
$$ \sup_{\substack t_n\in\tau_n
t_n\not\equiv 0}\dfrac{\|D^\alpha t_n\|_{q_1,q_2}}{\|t_n\|_{p_1,p_2}}\asymp I_\alpha n^{\alpha+\psi_1(p_1,p_2,q_1,q_2)}+n^{\alpha+\psi_2(p_1,p_2,q_1,q_2)}, $$
where
$$ I_\alpha=\begin{cases} \alpha,&0\leqslant\alpha\leqslant 1,1,&\alpha\geqslant 1, \end{cases} $$
and the functions $\psi_1$ and $\psi_2$ are given by an explicit formula. The asymptotic behaviour is with respect to $n$ for fixed $\alpha$, $p_1$, $p_2$, $q_1$ and $q_2$.

DOI: https://doi.org/10.4213/im223

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English version:
Izvestiya: Mathematics, 1998, 62:6, 1189–1206

Bibliographic databases:

MSC: 26A33, 41A17, 42A10
Received: 17.07.1997

Citation: A. I. Kozko, “Fractional derivatives and inequalities for trigonometric polynomials in spaces with asymmetric norms”, Izv. RAN. Ser. Mat., 62:6 (1998), 125–142; Izv. Math., 62:6 (1998), 1189–1206

Citation in format AMSBIB
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\by A.~I.~Kozko
\paper Fractional derivatives and inequalities for trigonometric polynomials in spaces with asymmetric norms
\jour Izv. RAN. Ser. Mat.
\yr 1998
\vol 62
\issue 6
\pages 125--142
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\zmath{https://zbmath.org/?q=an:0934.42001}
\transl
\jour Izv. Math.
\yr 1998
\vol 62
\issue 6
\pages 1189--1206
\crossref{https://doi.org/10.1070/im1998v062n06ABEH000223}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. I. Kozko, “On the order of the best approximation in spaces with asymmetric norm and sign-sensitive weight on classes of differentiable functions”, Izv. Math., 66:1 (2002), 103–131  mathnet  crossref  crossref  mathscinet  zmath
    2. K. V. Runovskii, “Multiplicator type operators and approximation of periodic functions of one variable by trigonometric polynomials”, Sb. Math., 212:2 (2021), 234–264  mathnet  crossref  crossref  mathscinet  isi  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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