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Zbl 0659.65004
Deslauriers, Gilles; Dubuc, Serge
Symmetric iterative interpolation processes. (English)
[J] Constructive Approximation 5, No.1, 49-68 (1989). ISSN 0176-4276; ISSN 1432-0940/e

A family of interpolation processes is introduced using two positive integral parameters: b a base and 2N an even number of moving nodes. Given y(n), n integer, the authors define $y(n+r/b)$, (r integer, $0<r<b)$ as the value of an interpolating Lagrange polynomial; the construction is iterated setting $y(j/b\sp{n+1})=\sum\sb{k}p\sb{j- kb}(k/b\sp n),$ the p's being finitely many parameters, j integer. An extension y(t) is thus obtained for the set of b-adic rational numbers. \par To obtain the properties of the process an associate function F(t) is defined satisfying the functional equation $F(t/b)=\sum\sb{n}F(n/b)F(t- n).$ The analysis of F(t) involves the trigonometric polynomials $P(\theta)=\sum\sb{k}F(k/b)e\sp{ik\theta}$ and the infinite matrix: $A=(F(k/b-j))\sb{-\infty <k<\infty,-\infty <j<\infty}$. F(t) is a continuous positive definite function; its order of regularity is precised. The function y(t) is defined as $y(t)=\sum\sb{n}y(n)F(t-n),$ and is proved to be uniformly continuous on any finite interval for all b, N and y(n). Error bounds and examples are given.
[A.de Castro]

MSC 2000:: *65D05 Interpolation (numerical methods)
65D10 Smoothing
41A05 Interpolation
42A05 Trigonometric polynomials

Keywords: curve fitting; Fourier transform; numerical examples; interpolation; Lagrange polynomial; trigonometric polynomials; Error bounds

Cited in: Zbl 1191.65011 Zbl 1124.65124 Zbl 1069.65020 Zbl 1047.65121 Zbl 0971.65008 Zbl 0902.41001 Zbl 0753.65007 Zbl 0763.42018

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