Converse Results

Jorge Bustamante
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Jorge Bustamante: Benemerita Universidad Autonoma de Puebla, Facultad de Ciencias Físico-Matemáticas

Chapter Chapter 6 in Bernstein Operators and Their Properties, 2017, pp 295-347 from Springer

Abstract: Abstract How much are the smoothness of a function and its rate of approximation by Bernstein operators related? What can be said about the structural properties of a function f ∈ C[0, 1], if some information on the behavior of the sequence { ∥ f − B n ( f) ∥ } is known? How do the singularities of a function influence the order of its approximation by Bernstein polynomials? There are several expressions concerning the connection between the rate of convergence of Bernstein polynomials and the smoothness of the function been approximated. In a previous chapter we have estimated the error in terms of different moduli of smoothness. By a converse result we mean an estimate of the moduli of smoothness in terms of the error of approximation. Converse results are much more difficult to prove than direct ones.

Date: 2017
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DOI: 10.1007/978-3-319-55402-0_6

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