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The aim of this note is to study the magnitude of the constants in the equivalence between the first and second order Ditzian-Totik moduli of smoothness and related K-functionals. Applications to some classic approximation operators are given.
In this work we discuss the rate of simultaneous approximation of Hölder continuous functions by Bernstein operators, measured by Hölder norms with different exponents. We extend the known results on this topic.
We represent new estimates of errors of quadrature formula, formula of numerical differentiation and approximation using Taylor polynomial. To measure the errors we apply representation of the remainder in Taylor formula by least concave majorant of the modulus of continuity of the n−th derivative of an n−times differentiable function. Our quantitative… (More)
In the present article we establish pointwise variant of E. V. Voronovskaja’s 1932 result, concerning the degree of approximation of Bernstein operator, applied to functions f ∈ C 3[0, 1].
We consider the question if lower estimates in terms of the second order Ditzian-Totik modulus are possible, when we measure the pointwise approximation of continuous function by Bernstein operator. In this case we confirm the conjecture made by Cao, Gonska and Kacsó. To prove this we first establish sharp upper and lower bounds for pointwise approximation… (More)