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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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