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Polynomial interpolation in several variables: lattices, differences, and ideals. (English) Zbl 1205.41004

Jetter, Kurt (ed.) et al., Topics in multivariate approximation and interpolation. Amsterdam: Elsevier (ISBN 978-0-444-51844-6). Studies in Computational Mathematics 12, 191-230 (2005).
Summary: When passing from one to several variables, the nature and structure of polynomial interpolation changes completely: the solvability of the interpolation problem with respect to a given finite dimensional polynomial space, like all polynomials of at most a certain total degree, depends not only on the number, but significantly on the geometry of the nodes. Thus the construction of interpolation sites suitable for a given space of polynomials or of appropriate interpolation spaces for a given set of nodes become challenging and nontrivial problems. The paper will review some of the basic constructions of interpolation lattices which emerge from the geometric characterization due to Chung and Yao. Depending on the structure of the interpolation problem, there are different representations of the interpolation polynomial and several formulas for the error of interpolation, reflecting the underlying point geometry and employing different types of differences. In addition, we point out the close relationship with constructive ideal theory and degree reducing interpolation, whose most prominent representer is the least interpolant, introduced by de Boor et al.
For the entire collection, see [K. Jetter (ed.), M. Buhmann (ed.), W. Haussmann (ed.), R. Schaback (ed.), J. Stoeckler (ed.), Topics in multivariate approximation and interpolation. Studies in Computational Mathematics 12. Amsterdam: Elsevier. (2005; Zbl 1205.41002)].
For the entire collection see [Zbl 1205.41002].

MSC:

41A05 Interpolation in approximation theory
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
41A63 Multidimensional problems

Citations:

Zbl 1205.41002

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