Interlineation and interflation functions of many variables (blending function interpolation) and economical algorithms in the approximation theory

Interflation of the function f(x(1),..., x(n)) of the n variables with help of the its traces (and traces of its derivatives of order <= N) on the M surfaces the dimension m is recovery (possible, exactly) f If m = 0 this is interpolation on M points (for n >= 1). If m = I (for n >= 2) it is interlineation (blending function interpolation) on M lines. In this paper the review of last achievements and some applications interflation, interlineation functions and blending approximation functions for construction the economical algorithms in the approximation theory is given.