On Optimal Recovery of Multivariate Periodic Functions

Abstract

Let X be a normed linear space of functions defined on the torus T^d: = [-π,π]^d and W c X. For a collection of points {x¹,…,x^k}c T^d and a mapping P_k(t₁,…,t_k) from R^k into a linear manifold in X of dimensions at most k, one can naturally consider recovering f∈w from its values f(x¹),…,f(x^k) by the element P_k(f(x¹),…f(x^k)).