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Interpolation through an iterative scheme. (English) Zbl 0615.65005

Let \(D_ n\) be the set of dyadic rationals \(m/2^ n\) where m is an arbitrary relative integer. Let y(t) be a function defined on \(D_ 0\). The author extends y(t) from \(D_ n\) to \(D_{n+1}\) by using the scheme \[ y(t)=[-y(t-3h)+9y(t-h)+9y(t+h)-y(t+3h)]/16,\quad with\quad h=2^{-n- 1}. \] The basic properties of this iterative scheme, which generates almost twice differentiable functions, are presented. A fundamental interpolating function F(t), arising from the sequence \(F(0)=1\) and \(F(n)=0\), is studied. The application to curve fitting is also considered.
Reviewer: L.Gatteschi

MSC:

65D05 Numerical interpolation
65D10 Numerical smoothing, curve fitting
41A05 Interpolation in approximation theory
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References:

[1] Akima, H., A new method of interpolation and smooth curve fitting based on Local procedures, J. Assoc. Comput. Mach., 17, 4, 589-602 (1970) · Zbl 0209.46805
[2] Lebesgue, H., Leçons sur les séries trigonométriques (1975), Albert Blanchard: Albert Blanchard Paris
[3] Zygmund, A., (Trigonometric Series, Vol. I (1968), Cambridge Univ. Press: Cambridge Univ. Press London)
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