On the Orders of Nonlinear Approximations for Classes of Functions of Given Form

Abstract

Suppose that Δ + s is the set of functions x: I → ℝ on a finite interval I such that the divided differences [x; t 0,..., t s ] of order s ∈ ℕ of these functions are nonnegative for all collections from (s +1) different points t 0,..., t s ∈ I. For all s ∈ ℕ and 1 ≤ p ≤ ∞, we establish exact orders of best approximations by splines with free nodes and rational functions in the metrics of L p for classes $$\Delta _ + ^s B_p : = \Delta _ + ^s \cap B_p$$ , where B p is the unit ball in L p . We also establish the asymptotics of pseudodimensional widths in L p of these classes of functions.