Information-based nonlinear approximation: an average case setting

Abstract

Nonlinear approximation (NA) has usually been studied under deterministic assumptions and complete information about the underlying functions. In the present paper we assume only partial information, e.g., function values at some points, and we are interested in the average case error and complexity of NA. We show that the problem can be essentially decomposed in two independent problems related to average case nonlinear (restricted) approximation from complete information, and to average case unrestricted approximation from partial information. The results are then applied to average case piecewise polynomial approximation on C ( [ 0 , 1 ] ) based on function values with respect to r-fold Wiener measure. In this case, to approximate with error ɛ it is necessary and sufficient to know the function values at Θ ɛ - 1 ln 1 / 2 ( 1 / ɛ ) 1 / ( r + 1 / 2 ) equidistant points and use Θ ɛ - 1 / ( r + 1 / 2 ) adaptively chosen break points in piecewise polynomial approximation.