Notes on Scattered-Data Radial-Function Interpolation

where α = (α1, . . . , αn) is an n-tuple of nonnegative integers and |α| = ∑n j=1 |αj |. If every polynomial p ∈ πm−1(R) is determined by its values on X, then we will say that the data set X is unisolvent (for πm−1(R )). This condition can also be rephrased in terms of matrices. Order the monomials x in some convenient way. Form the matrix P for which the rows are an x evaluated at xj , j = 1, . . . , N ; that is, the row corresponding to α is ( x1 · · · xN ). For example, if n = 2, m = 2 – so we are working in π1(R) – and N = 5, then P would be