Projection-net widths and lattice packings

Problems of approximation in a class of function spaces, including Sobolev spaces, by subspaces of finite-element type generated by translations of a lattice of given functions are considered. Widths that describe the approximation properties of such subspaces are defined, and their exact values are enumerated. Necessary and sufficient conditions are obtained for the optimality of subspaces on which these widths are realized. Criteria for the optimality of lattices in terms of the density of lattice packings of certain functions (for Sobolev spaces, of densities of packings by identical spheres) are established. Problems of comparison of the widths used in this article with the Kolmogorov widths of the same mean dimension are discussed.