Some Approximation Results on a Class of Szász-Mirakjan Operators Including Non-Negative Parameter Α

Abstract

Alternative proofs of the Weierstrass uniform approximation theorem have been provided by numerous mathematicians, including renowned ones. Among them, there was Bernstein that used a set of polynomials known as the Bernstein polynomials. Motivated by the advancements in computational disciplines, we propose a new type of Sz & aacute;sz-Mirakjan-Kantorovich operators that incorporate a shape parameter alpha. Certain shape-preserving properties, such as monotonicity and convexity, are achieved by computing the first and second order derivatives of the proposed operators. Certain approximation properties, including the statistical rate of convergence, are also obtained using a regular summability matrix. Finally, theoretical results are supported by illustrative graphics and numerical experiments using the Mathematica computer program. The operators defined in this paper may be used in computer and computational sciences, including in robotic manipulator control.