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Piecewise linear interpolation with nonequidistant nodes. (English) Zbl 0972.41007

Let \(\Delta_n:0= x_0<x_1< \cdots< x_n=1\) be a partition of the interval \([0,1]\) and let \(S_{\Delta_n}\) denote the positive linear operator associating with every \(f\in C[0,1]\) the piecewise linear and continuous function \(S_{\Delta_n}f\) interpolating \(f\) at the nodes \(x_0,\dots,x_n\). For two different choices of nodes the approximation error \(S_{\Delta_n}f-f\) is estimated pointwise and in the uniform norm on \([0,1]\) respectively in terms of the second order moduls of smoothness of \(f\) and in terms of the second order Ditzian-Totik modulus of smoothness with step-weight \(\varphi(x)= \sqrt{x(1-x)}\) respectively. Special attention is paid in both cases to the value of the constant in front of the modulus. The paper closes with a positive answer to a question posed by H. Gonska.

MSC:

41A10 Approximation by polynomials
41A15 Spline approximation
41A36 Approximation by positive operators
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[1] DOI: 10.1007/978-3-662-02888-9 · doi:10.1007/978-3-662-02888-9
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