On a Variational Norm Tailored to Variable-Basis Approximation Schemes

A variational norm associated with sets of computational units and used in function approximation, learning from data, and infinite-dimensional optimization is investigated. For sets Gk obtained by varying a vector y of parameters in a fixed-structure computational unit K(-,y) (e.g., the set of Gaussians with free centers and widths), upper and lower bounds on the G_K -variation norms of functions having certain integral representations are given, in terms of the £₁-norms of the weighting functions in such representations. Families of functions for which the two norms are equal are described.