Approximating fractional calculus operators with general analytic kernel by Stancu variant of modified Bernstein-Kantorovich operators

Abstract

The main aim of this paper is to approximate the fractional calculus (FC) operator with general analytic kernel by using auxiliary newly defined linear positive operators. For this purpose, we introduce the Stancu variant of modified Bernstein-Kantorovich operators and investigate their simultaneous approximation properties. Then we construct new operators by means of these auxiliary operators, and based on the obtained results, we prove the main theorems on the approximation of the general FC operators. We also obtain some quantitative estimates for this approximation in terms of modulus of continuity and Lipschitz class functions. Additionally, we exhibit our approximation results for the well-known FC operators such as Riemann-Liouville integral, Caputo derivative, Prabhakar integral, and Caputo-Prabhakar derivative.