Multivariate perpendicular interpolation.

*(English)*Zbl 0588.65004One of the really important problems of multivariate approximation theory is the one of interpolating to a large number of scattered data. Applications include the modeling of physical phenomena involving space and time coordinates and the computer aided design (CAD) of geometric objects. The author describes a nice geometric approach to multivariate interpolation. The paper is concerned with polyhedral domains D in \({\mathbb{R}}^ n\) that have been tesselated into n-dimensional simplices S. It is assumed that the values of all partial derivatives are given at the data points through the same order as the degree of global smoothness. The basic idea consists of constructing recursively the interpolant at an arbitrary point \(x\in S\) by forming

- the Taylor interpolant in the direction perpendicular to the (n-1)- dimensional faces of S (whence the name perpendicular interpolation in the title of the paper) in terms of barycentric coordinates \((x_ j)_{0\leq j\leq n}\) of \(x\in S\)

and then

- to apply a blending method which is based on extrapolation by means of a suitable nonnegative partition of unity in D.

The result of this recursive procedure is a piecewise rational function which has the prescribed global smoothness properties and interpolates to the original data. The perpendicular interpolants produced by the algorithm are local, i.e., their evaluation at any point \(y\in D\) requires only data on the simplex surrounding y. Explicit formulae of the geometric concepts introduced in the paper are given in the cases \(n=2\) and \(n=3\).

- the Taylor interpolant in the direction perpendicular to the (n-1)- dimensional faces of S (whence the name perpendicular interpolation in the title of the paper) in terms of barycentric coordinates \((x_ j)_{0\leq j\leq n}\) of \(x\in S\)

and then

- to apply a blending method which is based on extrapolation by means of a suitable nonnegative partition of unity in D.

The result of this recursive procedure is a piecewise rational function which has the prescribed global smoothness properties and interpolates to the original data. The perpendicular interpolants produced by the algorithm are local, i.e., their evaluation at any point \(y\in D\) requires only data on the simplex surrounding y. Explicit formulae of the geometric concepts introduced in the paper are given in the cases \(n=2\) and \(n=3\).

Reviewer: W.Schempp

##### MSC:

65D05 | Numerical interpolation |

41A05 | Interpolation in approximation theory |

41A63 | Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX) |