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Solvability of bivariate interpolation. II: Applications. (English) Zbl 0676.41004

This is the second of two paper [the first one will appear in Constructive Approximation] dealing with bivariate Lagrange interpolation by polynomials P(x,y) satisfying (1) \(P_{x^ sy^{\ell}}^{(j+\ell)}(z_ q)=c_{q,j,\ell},\) where Z is a set of distinct knots \(z_ q=(x_ q,y_ q)\in R^ 2,\) \(q=1,2,...,m\), \(P(x,y)=\sum a_{i,k}x^ iy^ k/i!k!,\) for (i,k) ranging over a given lower set of integral lattice points i,k\(\geq 0\), in \(R^ 2\). One associates with the problem (1) a multimatrix \(E=E\otimes...\otimes E_ m\) with 0-1 entries (1 at (j,\(\ell)\) means that the derivative \(\partial^{j+\ell}/\partial x^ j\partial x^{\ell}\) occurs in (1)). A result proved by J. C. Mairhuber in 1956 [see e.g. G. G. Lorentz, K. Jetter and S. D. Riemenschneider, Birkhoff Interpolation (1983; Zbl 0522.41001)] assert that problem (1) is solvable if and only if D(E,Z)\(\neq 0\) (D(E,Z) \(=\) the determinant of the system (1)) for every set Z of distinct knots in \(R^ 2\) if and only if E is an Abel matrix. The key idea of the authors is to relax this condition: Problem (1) is called almost regular (a.r.) if, as a polynomial in \(x_ 1,...,x_ m\), \(y_ 1,...,y_ m\), D(E,Z)\(\neq 0\) excepting a set of Lebesgue measure zero in \(R^{2m}\). Otherwise, D(E,Z)\(\equiv 0\) and the problem (1) is called singular. The a.r. regularity of the multi-matrix E the problem (1) makes for the system Z of knots solvable (this can be done with probability one). In many particular cases the authors find explicitely the set Z giving applications to finite elements methods. Also the singularity of E warns the numerical analyst that the interpolation schemes have to be avoided.
Reviewer: C.Mustăţa

MSC:

41A05 Interpolation in approximation theory
41A63 Multidimensional problems

Citations:

Zbl 0522.41001
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