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On one method of numerical integration. (English) Zbl 0753.65020

By using the Lienhard interpolation method, the authors construct an interpolation curve passing through the points \(P_ j(x_ j,f(x_ j))\), (\(j=1,2,\dots,n+1\)), where \(f\in C[a,b]\) and \(x_ j=a+(j-1)h\), \(j=1,2,\dots,n+1\) and \(h=(b-a)/n\). The arcs \(P_ jP_{j+1}\) are parametrized by means of some given polynomials of third degree, depending on a parameter \(t\in [-1,1]\). They prove the uniform convergence of the sequence of Lienhard interpolants to the corresponding continuous function \(f\) on \([a,b]\).
In the second part of the paper a numerical integration rule of functions based on the Lienhard interpolation method is derived. Numerous simple examples are given. The paper also contains two programs in BASIC, useful for constructing approximating Lienhard curves.

MSC:

65D32 Numerical quadrature and cubature formulas
65D05 Numerical interpolation
41A55 Approximate quadratures
41A05 Interpolation in approximation theory
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References:

[1] H. Lienhard: Interpolation von Funktionswerten bei nuraerischen Bahnsteuerungen. Undated publication of CONTRAVES AG, Zurich. · Zbl 0147.35803
[2] J. Matušů: The Lienhard interpolation method and some of its generalization. (in Czech). Act. Polytechnica - Práce ČVUT, Prague 3 (IV, 2), 1978.
[3] J. Matušů J. Novák: Constructions of interpolation curves from given supporting elements (I). Aplikace matematiky 30 (1985), 4, Prague. · Zbl 0605.65006
[4] J. Matušů J. Novák: Constructions of interpolation curves from given supporting elements (II). Aplikace matematiky 31 (1986), 2, Prague. · Zbl 0628.65004
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