Convergence analysis of Kantorovich-type operators in variable exponent Sobolev spaces

Abstract In this work, we establish convergence results in variable exponent Sobolev spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>W</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mo>⋅</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="double-struck">R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> W^{1,p(\,{\cdot}\,)}(\mathbb{R}) , where the exponent function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> </m:mrow> </m:math> p\colon\mathbb{R}\to\mathbb{R} satisfies <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:msup> <m:mi>p</m:mi> <m:mo>−</m:mo> </m:msup> <m:mo>≤</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>≤</m:mo> <m:msup> <m:mi>p</m:mi> <m:mo>+</m:mo> </m:msup> <m:mo>&lt;</m:mo> <m:mrow> <m:mo>+</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:mrow> </m:math> 1\leq p^{-}\leq p(x)\leq p^{+}&lt;+\infty for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> </m:math> x\in\mathbb{R} , for Kantorovich-type operators activated by kernel functions. Furthermore, we prove uniform convergence for functions belonging to appropriate function spaces.