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Bernstein-Schoenberg operator with knots at the $$p$$-integers. (English) Zbl 1255.41003
Summary: We consider a special knot sequence $$u_{i+1}=qu_{i}+1$$ and define a one parameter family of Bernstein-Schoenberg operators. We prove that this operator converges to $$f$$ uniformly for all $$f$$ in $$C[0,1]$$. This operator also inherits the geometric properties of the classical Bernstein-Schoenberg operator. Moreover we show that the error function $$E_{m,n}$$ has a particular symmetry property, that is $$E_{m,n}(f;x;q)=E_{m,n}(f;1 - x,1/q)$$ provided that $$f$$ is symmetric on $$[0,1]$$.

##### MSC:
 41A35 Approximation by operators (in particular, by integral operators) 41A15 Spline approximation
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##### References:
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