Stefano De Marchi: Aldaz-Kounchev-Render operators: approximation properties and their extension to higher dimensions

The Aldaz-Kounchev-Render (AKR) operator was introduced in [3] and afterwards investigated in several papers (cf. [1]). We improve some existing quantitative results concerning these approximation properties. Moreover, we describe classes of functions for which these operators approximate better than the classical Bernstein operators and classes of functions for which Bernstein operators approximate better than AKR operators. The new results, in particular involving monotonic convergence and Voronovskaja type formulas, are then extended to the bivariate case on the square [0,1]^2 and compared with other existing results. We also discuss the Aldaz-Kounchev-Render operators on a multidimensional simplex [2]. In the case of the unit simplex of R m these operators preserve the functions 1, x_j^1 , . . . , x_j^m, j a positive integer. The Voronovskaja type formula is also conjectured. References [1] A.M.Acu, S. De Marchi, I. Rasa, Aldaz–Kounchev–Render operators and their approximation properties, Results Math 78, 21 (2023). [2] A.M.Acu, S. De Marchi, I. Rasa, Aldaz-Kounchev-Render operators on simplices , J. Math. Anal. Appl. 533 (2), (2024). [3] J.M. Aldaz, O. Kounchev, H. Render, Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces, Numer. Math., 114(1) (2009), 1–25. The zoom link to the talk is: https://us06web.zoom.us/j/83312342576?pwd=muM3vzOQ4nIezupfbcmNRI28855u9x.1