Revisiting Kantorovich Operators in Lebesgue Spaces Maximillian Ventura Obie Welly, Erick Angga Taebenu, Reinhart Gunadi, Denny Ivanal Hakim
Institut Teknologi Bandung
Abstract
According to the Weierstrass Approximation Theorem, any continuous function on the closed and bounded interval can be approximated by polynomials. A constructive proof of this theorem uses the so-called Bernstein polynomials. For the approximation of integrable functions, we may consider Kantorovich operators as certain modifications for Bernstein polynomials. In this paper, we investigate the behavior of Kantorovich operators in Lebesgue spaces. We first give an alternative proof of the uniform boundedness of Kantorovich operators in Lebesgue spaces by using the Riesz-Thorin Interpolation Theorem. In addition, we examine the convergence of Kantorovich operators in the space of essentially bounded functions. We also consider the rate of convergence of Kantorovich operators in some subspaces of Lebesgue spaces.
Keywords: Kantorovich operators, Lebesgue spaces, interpolation of linear operators
