Quantitative Estimates for Positive Linear Operators in terms of the Usual Second Modulus. (11th May 2015)

• Record Type: Journal Article
• Authors: Adell, José A.; Lekuona, A.
• Other Names: Pokorny Milan Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2015(2015)
• Publication Date: 2015-05-11
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2015/915358
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : We give accurate estimates of the constantsC n ( A ( I ), x ) appearing in direct inequalities of the form| L n f ( x ) - f ( x ) | ≤ C n ( A ( I ), x ) ω 2 f ; σ ( x ) / n, f ∈ A ( I ), x ∈ I, and n = 1, 2, …, whereL n is a positive linear operator reproducing linear functions and acting on real functionsf defined on the intervalI, A ( I ) is a certain subset of such functions, ω 2 ( f ; · ) is the usual second modulus off, andσ ( x ) is an appropriate weight function. We show that the size of the constantsC n ( A ( I ), x ) mainly depends on the degree of smoothness of the functions in the setA ( I ) and on the distance from the pointx to the boundary ofI . We give a closed form expression for the best constant whenA ( I ) is a certain set of continuous piecewise linear functions. As illustrative examples, the Szàsz-Mirakyan operators and the Bernstein polynomials are discussed.