On the Sets of Convergence for Sequences of the ð ‘ž -Bernstein Polynomials with ð ‘ž > 1

Sofiya Ostrovska and Ahmet Yaşar Özban

Abstract and Applied Analysis, 2012, vol. 2012, 1-19

Abstract:

The aim of this paper is to present new results related to the convergence of the sequence of the ð ‘ž -Bernstein polynomials { ð µ ð ‘› , ð ‘ž ( ð ‘“ ; ð ‘¥ ) } in the case ð ‘ž > 1 , where ð ‘“ is a continuous function on [ 0 , 1 ] . It is shown that the polynomials converge to ð ‘“ uniformly on the time scale ð • ð ‘ž = { ð ‘ž − ð ‘— } ∞ ð ‘— = 0 ∪ { 0 } , and that this result is sharp in the sense that the sequence { ð µ ð ‘› , ð ‘ž ( ð ‘“ ; ð ‘¥ ) } ∞ ð ‘› = 1 may be divergent for all ð ‘¥ ∈ ð ‘… ⧵ ð • ð ‘ž . Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:185948

DOI: 10.1155/2012/185948

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