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Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2016, Volume 16, Issue 3, Pages 322–330 (Mi isu651)  

Mathematics

On convergence of Bernstein–Kantorovich operators sequence in variable exponent Lebesgue spaces

T. N. Shakh-Emirov

Daghestan Scientific Centre of RAS, 45, Gadgieva st., 367000, Makhachkala, Republic of Dagestan, Russia

Abstract: Let $E=[0,1]$ and let a function $p(x)\ge1$ be measurable and essentially bounded on $E$. We denote by $L^{p(x)}(E)$ the set of measurable function $f$ on $E$ for which $\int_{E}|f(x)|^{p(x)}dx<\infty$. The convergence of a sequence of operators of Bernstein–Kantorovich $\{K_n(f,x)\}_{n=1}^\infty$ to the function $f$ in Lebesgue spaces with variable exponent $L^{p(x)}(E)$ is studied. The conditions on the variable exponent at which this sequence is uniformly bounded in these spaces are obtained and, as a corollary, it is shown that if $n\to\infty$ then $K_n(f,x)$ converges to function $f$ in the metric of space $L^{p(x)}(E)$ defined by the norm $\|f\|_{p(\cdot)}=\|f\|_{p(\cdot)}(E)=\inf\{\alpha>0:\quad\int\limits_E|\frac{f(x)}\alpha|^{p(x)}dx\le1\}$.

Key words: Lebesgue spaces with variable exponent, Bernstein–Kantorovich operators, Bernstein polynomials.

DOI: https://doi.org/10.18500/1816-9791-2016-16-3-322-330

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Bibliographic databases:

UDC: 517.51

Citation: T. N. Shakh-Emirov, “On convergence of Bernstein–Kantorovich operators sequence in variable exponent Lebesgue spaces”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 16:3 (2016), 322–330

Citation in format AMSBIB
\Bibitem{Sha16}
\by T.~N.~Shakh-Emirov
\paper On convergence of Bernstein--Kantorovich operators sequence in variable exponent Lebesgue spaces
\jour Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform.
\yr 2016
\vol 16
\issue 3
\pages 322--330
\mathnet{http://mi.mathnet.ru/isu651}
\crossref{https://doi.org/10.18500/1816-9791-2016-16-3-322-330}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3557760}
\elib{http://elibrary.ru/item.asp?id=26702022}


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  • Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика
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