GENERAL INTERPOLATION ON THE LATTICE HZ(S) - COMPACTLY SUPPORTED FUNDAMENTAL-SOLUTIONS

In this paper we combine an earlier method developed with K. Jetter on general cardinal interpolation with constructions of compactly suppor ted solutions for cardinal interpolation to gain compactly supported f undamental solutions for the general interpolation problem. The genera l interpolation problem admits the interpolation of the functional and derivative values under very weak restrictions on the derivatives to be interpolated. In the univariate case, some known general constructi ons of compactly supported fundamental solutions for cardinal interpol ation are discussed together with algorithms for their construction th at make use of MAPLE. Another construction based on finite decompositi on and reconstruction for spline spaces is also provided. Ideas used i n the latter construction are lifted to provide a general construction of compactly supported fundamental solutions for cardinal interpolati on in the multivariate case. Examples are provided, several in the con text of some general interpolation problem to illustrate how easy is t he transition from cardinal interpolation to general interpolation.