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Multivariate interpolation. (English) Zbl 0566.41001

Rational approximation and interpolation, Proc. Conf., Tampa/Fla. 1983, Lect. Notes Math. 1105, 136-144 (1984).
[For the entire collection see Zbl 0544.00011.]
Let E be an interpolation multimatrix consisting of matrices \(E_ q\), \(q=1,...,m\) with elements \(e_{q,i,k}+1\) or \(=0\). We consider the interpolation problem (*) \((\partial^{i+k}/\partial x^ i\partial y^ k)P|_{(x,y)=(x_ q,y_ q)}=\) given data, \(e_{q,i,k}=1\), \(q=1,...,m\), for polynomials of the form \(P(x,y)=\sum_{(i,k)\in S}a_{i,k}x^ iy^ k\). The problem is solvable for all selections of knots \((x_ q,y_ q)\) and all given data if and only if E is an Abel matrix, that is, if each derivative with (i,k)\(\in S\) appears exactly once in the equations (*). Since Lagrange matrices are not Abel matrices, this is a generalization of Mairhuber’s theorem.

MSC:

41A05 Interpolation in approximation theory

Citations:

Zbl 0544.00011