Optimal recovery of linear functionals of Peano type through data on manifolds

The usual approach to optimal recovery is based on the information obtained through functionals, e.g., point evaluation, integrals, etc. We introduce a new concept in the multivariate optimal recovery which is based on information obtained through operators. In particular, we use operators which are given by the restriction of a function to a union of manifolds of dimension n − 1. The motivation for introducing this new concept is the complete analogy of our results with the results of Schoenberg and others in the area of one-dimensional optimal recovery.