Approximation by Max-Product Operators of Kantorovich Type

The main goal of this survey is to describe the results of the present authors concerning approximation properties of various max-product Kantorovich operators, fulfilling thus this gap in their very recent research monograph (Bede et al., Approximation by Max-product type operators. Springer, New York, 2016). Section 1 contains a short introduction in the topic. In Sect. 2, after presenting some general results, we state approximation results including upper estimates, direct and inverse results, localization results and shape preserving results, for the max-product: Bernstein–Kantorovich operators, truncated and non-truncated Favard–Szász-Mirakjan—Kantorovich operators, truncated and non-truncated Baskakov–Kantorovich operators, Meyer–König-Zeller–Kantorovich operators, Hermite–Fejér–Kantorovich operators based on the Chebyshev knots of first kind, discrete Picard–Kantorovich operators, discrete Weierstrass–Kantorovich operators and discrete Poisson–Cauchy–Kantorovich operators. All these approximation properties are deduced directly from the approximation properties of their corresponding usual max-product operators. Section 3 presents the approximation properties with quantitative estimates in the Lp-norm, 1 ≤ p ≤ +∞, for the Kantorovich variant of the truncated max-product sampling operators based on the Fejér kernel. In Sect. 4, we introduce and study the approximation properties in Lp-spaces, 1 ≤ p ≤ +∞ for truncated max-product Kantorovich operators based on generalized type kernels depending on two functions φ and ψ satisfying a set of suitable conditions. The goal of Sect. 5 is to present approximation in Lp, 1 ≤ p ≤ +∞, by sampling max-product Kantorovich operators based on generalized kernels, not necessarily with bounded support, or generated by sigmoidal functions. Several types of kernels for which the theory applies and possible extensions and applications to higher dimensions are presented. Finally, some new directions for future researches are presented, including applications to learning theory.