Geodesic
On convergence of Bernstein–Kantorovich operators sequence in variable exponent Lebesgue spaces
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 3, pp. 322-330 
T. N. Shakh-Emirov. On convergence of Bernstein–Kantorovich operators sequence in variable exponent Lebesgue spaces. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 3, pp. 322-330. http://geodesic.mathdoc.fr/item/ISU_2016_16_3_a9/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\|f\|_{p(\cdot)}=\|f\|_{p(\cdot)}(E)=\inf\left\{\alpha>0:\quad\int\limits_E\left|\frac{f(x)}\alpha\right|^{p(x)}dx\le1\right\}$.

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