ABSTRACT
In Part I we will only consider interpolation at all of the grid points of a rectangular grid. This problem arises frequently in applications. Suppose that we are given
(Li) The nm points in the #î/-plane,
(1.2) are called the grid points of the (axes parallel) rectangle,
(1.3) This rectangle is subdivided into (n — l)(ra — 1) subrectangles,
(1.4) with vertices,
(1.5)
The n + m grid lines are given by
(1.6)
In this way, the numbers x\, • • •, xn, y\, • • •, ym with property (1.1), define a rectangular grid and fix the arrangement of the subrectangles Rij. (A given skew grid or lattice is always the unique, invertible affine image of a rectangular grid.)
(1.7)
Typically, we will postulate F to be of a certain form so that then, finding the interpolant amounts to finding the coefficients of certain basis functions. Thus, the the interpolation nodes coincide with the grid points. (Sometimes we will also allow given values for certain partial derivatives.)
(1.8)
is available, then clearly the product,
solves the interpolation problem (1.7). Unfortunately, as we shall see in the examples to follow, simply con-
structed cardinal functions do not always produce visually pleasing and smooth interpolation surfaces F = F(x,y).
