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

Suppose that Δ is the set of functions x: I → ℝ on a finite interval I such that the divided differences [ x; t₀,..., t _s] of order _s ∈ ℕ of these functions are nonnegative for all collections from ( s +1) different points t₀,..., 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. [ABSTRACT FROM AUTHOR]