Cite this article as:
Shakh-Emirov T. N. On Convergence of Bernstein – Kantorovich Operators sequence in Variable Exponent Lebesgue Spaces. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2016, vol. 16, iss. 3, pp. 322-330. DOI: https://doi.org/10.18500/1816-9791-2016-16-3-322-330
On Convergence of Bernstein – Kantorovich Operators sequence in Variable Exponent Lebesgue Spaces
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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