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The saturation of convergence on the interval [0,1] for the $q$-Bernstein polynomials in the case $q>1$. (English) Zbl 1236.41011
Summary: We consider saturation of convergence on the interval $[0,1]$ for the $q$-Bernstein polynomials of a continuous function $f$ for arbitrary fixed $q>1$. We show that the rate of uniform convergence on $[0,1]$ is $o(q ^{- n})$ if and only if $f$ is linear. The result is sharp in the following sense: it ceases to be true if we replace “$o$” by “$O$”.

41A10Approximation by polynomials
33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
Full Text: DOI
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