On the Generalized Szász–Mirakyan Operators

• Published in: Resultate der Mathematik Vol. 65; no. 3-4; pp. 441 - 452
• Main Authors: Aral, A., Inoan, D., Raşa, I.
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01.06.2014
• Subjects: Mathematics; Mathematics and Statistics; weighted modulus of continuity; 41A36; 41A25; weighted approximation; Generalized Szász–Mirakyan operators; shape preserving properties
• ISSN: 1422-6383; 1420-9012
• DOI: 10.1007/s00025-013-0356-0

In this paper, we construct sequences of Szász–Mirakyan operators which are based on a function ρ . This function not only characterizes the operators but also characterizes the Korovkin set 1 , ρ , ρ 2 in a weighted function space. We give theorems about convergence of these operators to the identity operator on weighted spaces which are constructed using the function ρ and which are subspaces of the space of continuous functions on R + . We give quantitative type theorems in order to obtain the degree of weighted convergence with the help of a weighted modulus of continuity constructed using the function ρ . Further, we prove some shape-preserving properties of the operators such as the ρ -convexity and the monotonicity. Our results generalize the corresponding ones for the classical Szász operators.