Towards the best constant in front of the Ditzian-Totik modulus of smoothness

• Published in: Journal of inequalities and applications 2016-05, Vol.2016 (1), p.1-17, Article 137
• Main Authors: Adell, José A, Lekuona, Alberto
• Format: Article
• Language: English
• Subjects: Analysis; Applications of Mathematics; best constant; Boundary conditions; Constants; direct inequality; Ditzian-Totik modulus of smoothness; Inequalities; Linear operators; Mathematics; Mathematics and Statistics; Operators; positive linear operator; Smoothness; Szàsz-Mirakyan operator; Texts; Weierstrass operator; Weight function
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-016-1078-0

We give accurate estimates for the constants K ( A ( I ) , n , x ) = sup f ∈ A ( I ) | L n f ( x ) − f ( x ) | ω σ 2 ( f ; 1 / n ) , x ∈ I , n = 1 , 2 , … , where I = R or I = [ 0 , ∞ ) , L n is a positive linear operator acting on real functions f defined on the interval I , A ( I ) is a certain subset of such function, and ω σ 2 ( f ; ⋅ ) is the Ditzian-Totik modulus of smoothness of f with weight function σ . This is done under the assumption that σ is concave and satisfies some simple boundary conditions at the endpoint of I , if any. Two illustrative examples closely connected are discussed, namely, Weierstrass and Szàsz-Mirakyan operators. In the first case, which involves the usual second modulus, we obtain the exact constants when A ( R ) is the set of convex functions or a suitable set of continuous piecewise linear functions.