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The approximation of logarithmic function by q-Bernstein polynomials in the case q > 1

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Abstract

Since in the case q > 1, q-Bernstein polynomials are not positive linear operators on C[0,1], the study of their approximation properties is essentially more difficult than that for 0<q<1. Despite the intensive research conducted in the area lately, the problem of describing the class of functions in C[0,1] uniformly approximated by their q-Bernstein polynomials (q > 1) remains open. It is known that the approximation occurs for functions admit ting an analytic continuation into a disc {z:|z| < R}, R > 1. For functions without such an assumption, no general results on approximation are available. In this paper, it is shown that the function f(x) = ln (x + a), a > 0, is uniformly approximated by its q-Bernstein polynomials (q > 1) on the interval [0,1] if and only if a ≥ 1.

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Correspondence to Sofiya Ostrovska.

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Ostrovska, S. The approximation of logarithmic function by q-Bernstein polynomials in the case q > 1. Numer Algor 44, 69–82 (2007). https://doi.org/10.1007/s11075-007-9081-7

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  • DOI: https://doi.org/10.1007/s11075-007-9081-7

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