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Symmetric iterative interpolation processes. (English) Zbl 0659.65004

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 n+1 )= k p j-kb (k/b 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.

To obtain the properties of the process an associate function F(t) is defined satisfying the functional equation F(t/b)= n F(n/b)F(t-n)· The analysis of F(t) involves the trigonometric polynomials P(θ)= k F(k/b)e ikθ and the infinite matrix: A=(F(k/b-j)) -<k<,-<j< . F(t) is a continuous positive definite function; its order of regularity is precised. The function y(t) is defined as y(t)= 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.

Reviewer: A.de Castro

65D05Interpolation (numerical methods)
65D10Smoothing, curve fitting
41A05Interpolation (approximations and expansions)
42A05Trigonometric polynomials, inequalities, extremal problems
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