A comprehensive study on the shape properties of Kantorovich type Schurer operators equipped with shape parameter λ - Turhan, Nezihe; Özger, Faruk; Ödemiş Özger, Zeynep (Expert systems with applications, 2025)
This study explores the shape-preserving characteristics of the Kantorovich variant of λ−Schurer operators which are a modified version of the classical Kantorovich-type Schurer operators enhanced by the introduction of a shape parameter λ∈−1,1. The underlying objective of this study is to analyze how these operators retain the intrinsic geometric characteristics of the functions they approximate, a quality essential in applications such as computer graphics, signal processing, geometric modeling, robotics, and so forth. To accomplish this task, we begin by representing the operators in question as a sum of the classical Kantorovich-type Schurer operators and an additional term involving the integral of first-order divided differences of the function ℓ, where this additional term is scaled by the shape parameter λ. Using this formulation and the fundamental properties of divided differences, we then investigate the shape-preserving characteristics of these operators, including linearity, positivity, and, in particular, their ability to maintain monotonicity and convexity in relation to the function ℓ. The outcomes of this study show that the operators fully preserve monotonicity over the interval 0,1 for all λ∈−1,1, but fail to consistently preserve convexity for certain values of λ within the same range. We support this conclusion with counterexamples and provide an adjusted result on convexity preservation for a particular class of functions when λ is chosen from the interval 0,1. We conclude our analysis with a section focusing on the convexity-preserving comparison of operators characterized by the shape parameter λ. •Integration of shape parameter λ into Kantorovich type Schurer operators for enhanced adaptability.•Operators preserve monotonicity across all λ values and partially preserve convexity.•Relevant for geometric modeling, computer graphics, and signal processing.