On the Monotonicity of Positive Linear Operators

✍ Scribed by M.Kazim Khan; B. Della Vecchia; A. Fassih

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 377 KB
• Volume: 92
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.3113

✦ Synopsis

We provide sufficient conditions for a sequence of positive linear approximation operators, L n ( f, x), converging to f (x) from above to imply the convexity of f. We show that, for the convolution operators of Feller type, K n ( f, x), generated by a sequence of iid random variables taking values in an interval I, having a finite moment generating function, the inequalities K n ( f, x) f (x) (x # I, n 1) are necessary and sufficient conditions for f to be convex. This provides a converse of a well-known result of R. A. Khan (Acta. Math. Acad. Sci. Hungar. 39 (1980), 193 203). It contains, as a special case, the corresponding result for the Bernstein polynomials and extends two results obtained for bounded continuous functions by Horova for Sza sz and Baskakov operators. As examples, similar results are also provided for the beta, Meyer-Ko nig Zeller, Picard, and Bleiman, Butzer, and Hahn operators. 1998 Academic Press 1. INTRODUCTION Let I be an interval and, for each x # I, let [+ n, x , n=1, 2, ...] be a sequence of finite measures concentrating on I. Let g(t) be a non-negative continuous function increasing to infinity as t Ä \ . We define D g (I ) to Article No. AT963113 22