Scattered data points best interpolation as a problem of the best recovery in the sense of Sard

Abstract

The problem of the best recovery in the sense of Sard of a linear functional Lf on the basis of information T(f) = {Ljf,j = 1, 2,... N} is studied. It is shown that in the class of bivariate functions with restricted (n,m) -derivative, known on the (n,m)-grid lines, the problem of the best recovery of a linear functional leads to the best approximation of L(KnKm) in the space S = span Lj(Kn_Km), j=1; 2,...N}, where Kn(x,t) = K(x,t)- Lxn(K(.,t):x) is the difference between the truncated power kernel K(x,t) = (x-t)n-1+ =(n-1)! and its Lagrange interpolation formula. In particular, the best recovery of a bivariate function is considered, if scattered data points and blending grid are given. An algorithm is designed and realized using the software product MATLAB.