Journal
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS
Volume 36, Issue 5, Pages 1615-1629Publisher
ROCKY MT MATH CONSORTIUM
DOI: 10.1216/rmjm/1181069386
Keywords
Bernstein polynomials; q-integers; q-binomial coefficients; convergence
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Let R-n(f,q;x) : C[0, 1] -> C[0, 1] be q-analogues of the Bernstein operators defined by Lupas in 1987. If q = 1, then R-n (f, 1; x) are classical Bernstein polynomials. For q not equal 1, the operators R-n (f, q; x) are rational functions rather than polynomials. The paper deals with convergence properties of the sequence {R-n (f, q; x)}. It is proved that {R-n (f, q(n); x)} converges uniformly to f(x) for any f(x) is an element of C[0, 1] if and only if q(n) -> 1. In the case q > 0, q not equal 1 being fixed the sequence I R. (f, q; x) I converges uniformly to f(x) is an element of C[0, 1] if and only if f(x) is linear.
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