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

The problem of the best recovery in the sense of Sard of a linear functional Lf on the basis of information T (f) = fLjf; j = 1; 2; : : : ; Ng 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 = spanfLj(Kn Km); j = 1; 2; : : : ; Ng, where Kn(x; t) = K(x; t) Lxn(K(:; t); x) is the dierence 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.