Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Shakh-Emirov T. N. On Convergence of Bernstein – Kantorovich Operators sequence in Variable Exponent Lebesgue Spaces. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 3, pp. 322-330. DOI: 10.18500/1816-9791-2016-16-3-322-330, EDN: WMIIIL

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On Convergence of Bernstein – Kantorovich Operators sequence in Variable Exponent Lebesgue Spaces

Shakh-Emirov T. N., Daghestan Scientific Centre of Russian Academy of Sciences

Let E = [0, 1] and let a function p(x) > 1 be measurable and essentially bounded on E. We denote by L p(x) (E) the set of measurable function f on E for which R E |f(x)| p(x) dx < ∞. The convergence of a sequence of operators of Bernstein – Kantorovich {Kn(f, x)} ∞n=1 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 → ∞ then Kn(f, x) converges to function f in the metric of space L p(x) (E) defined by the norm.

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