Spline approximation and optimal recovery of operators

The problem of optimal recovery, on the basis of exact or erroneous information, of symmetry-preserving operators on sets of elements of convolution type is solved. Using the information operator and a generating kernel, an approximation apparatus is constructed, called information-kernel splines. In particular cases, it coincides with sets of polynomial splines in one or several variables. Interpolation and smoothing are solvable for it.