Abstract
In this chapter, we consider the problem of interpolation of values of a function and its partial derivatives by multivariate polynomials from a certain finite-dimensional space. The interpolation problem consists of the following components:
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a)
the space of polynomials
$$\pi (S) = \{ P:P(x) = P({x_1},...,{x_k}) = \sum\limits_{\alpha = ({\alpha _1},...,{\alpha _k}) \in S} {{a_\alpha }x_1^{{\alpha _k}}} \} $$where S ⊂ ℤ k+ is a finite normal set, i.e., α ∈ S, β ∈ ℤ k+ and β ≤ α imply β ∈ S;
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b)
collection of sets
$$\mathcal{H} = \{ {H_v}\} _{v = 1}^s,\;{H_v} \subset {\text{ }}{\mathbb{R}^k},\;\left| \mathcal{H} \right|: = \sum\limits_{v = 1}^s {\left| {{H_v}} \right|} = \left| S \right| = \dim \pi (S);$$((14.1.1)) -
c)
set of nodes
$$Z = \{ {z_v}\} _{v = 1}^s,\;{z_v} \subset {\text{ }}{\mathbb{R}^k},\;{z_v} \ne {z_\mu }\;for{\text{ }}v \ne \mu ,\;v,\mu = 1,...,s.$$((14.1.2))
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© 1993 Springer Science+Business Media Dordrecht
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Bojanov, B.D., Hakopian, H.A., Sahakian, A.A. (1993). Multivariate Pointwise Interpolation. In: Spline Functions and Multivariate Interpolations. Mathematics and Its Applications, vol 248. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8169-1_14
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DOI: https://doi.org/10.1007/978-94-015-8169-1_14
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4259-0
Online ISBN: 978-94-015-8169-1
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