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Lidstone interpolation. III: Several variables. (English) Zbl 1507.41012

Univariate polynomials can be identified uniquely by Hermite interpolation conditions at two points in the complex plane, the arguments zero and one, say. The order of these Hermite conditions has to be sufficiently large, and it is suitable to demand the conditions at the two points with only even-order derivatives. In this paper, this theory is generalised to \(n\) dimensions, and the points at 0 and 1 are replaced by suitable unit vectors in \(n\) variables. A canonical representation using this Hermite interpolation information is an expansion into linear combinations of so-called Lidstone polynomials that replace the well-known Lagrange functions in Lagrange and Hermite interpolation of the classical form.

MSC:

41A63 Multidimensional problems
32A08 Polynomials and rational functions of several complex variables
32A15 Entire functions of several complex variables
41A05 Interpolation in approximation theory
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
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References:

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