On the best recovery of a linear functional in a certain class of bivariate functions

Abstract

Let f(x, y) be a bivariate function in a Hilbert space H. In the paper the problem of the best recovery in the sense of Sard of a linear functional Lf on the basis of information is studied. It is shown that in the class of 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(K n K m) in the space where is the difference between the truncated power kernel and its Lagrange interpolation formula. The analog of Shoenberg's Theorem for bivariate functions is proved . In particular, the best recovery of a bivariate function is considered. An algorithm is designed and realized using the software product MATLAB.