Approximation by Mixed Operators of Max-Product–Choquet Type

The main aim of this chapter is to introduce several mixed operators between Choquet integral operators and max-product operators and to study their approximation, shape preserving, and localization properties. Section 2 contains some preliminaries on the Choquet integral. In Sect. 3, we obtain quantitative estimates in uniform and pointwise approximation for the following mixed type operators: max-product Bernstein–Kantorovich–Choquet operator, max-product Szász–Mirakjan–Kantorovich–Choquet operators, nontruncated and truncated cases, and max-product Baskakov–Kantorovich–Choquet operators, nontruncated and truncated cases. We show that for large classes of functions, the max-product Bernstein–Kantorovich–Choquet operators approximate better than their classical correspondents, and we construct new max-product Szász–Mirakjan–Kantorovich–Choquet and max-product Baskakov–Kantorovich–Choquet operators, which approximate uniformly f in each compact subinterval of [0, +∞) with the order ω 1 ( f ; λ n ) $$\omega _{1}(f; \sqrt {\lambda _{n}})$$ , where arbitrary fast. Also, shape preserving and localization results for max-product Bernstein–Kantorovich–Choquet operators are obtained. Section 4 contains quantitative approximation results for discrete max-product Picard–Kantorovich–Choquet, discrete max-product Gauss–Weierstrass–Kantorovich–Choquet operators, and discrete max-product Poisson–Cauchy–Kantorovich–Choquet operators. Section 5 deals with the approximation properties of the max-product Kantorovich–Choquet operators based on (φ, ψ)-kernels. It is worth to mention that with respect to their max-product correspondents, while they keep their good properties, the mixed max-product Choquet operators present, in addition, the advantage of a great flexibility by the many possible choices for the families of set functions used in their definitions. The results obtained present potential applications in sampling theory, neural networks, and learning theory.