The Second Ditzian-Totik modulus revisited: refined estimates for positive linear operators

We give accurate estimates for the constants K(A(I), n, x) = sup f ∈A(I) |L n f (x)-f (x)| ω 2 σ (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 2 σ (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.

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The Second Ditzian-Totik modulus revisited: refined estimates for positive linear operators

Cited by 3 publications

References 22 publications

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Upper Error Estimates of Bernstein Operators

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