Uniform approximation of piecewise r-smooth and globally continuous functions

We study the uniform (Chebyshev) approximation of continuous and piecewise r-smooth (r ≥ 2) functions f : [0, T] → ℝ with a finite number of singular points. The approximation algorithms use only n function values at adaptively or nonadaptively chosen points. We construct a nonadaptive algorithm A_r,n^non that, for the functions with at most one singular point, enjoys the best possible convergence rate n^-r. This is in sharp contrast to results concerning discontinuous functions. For r ≥ 3, this optimal rate of convergence holds only in the asymptotic sense, i.e., it occurs only for sufficiently large n that depends on f in a way that is practically impossible to verify. However, it is enough to modify A_r,n ^non by using (r + 1) ⌊(r - 1)/2⌋ extra function evaluations to obtain an adaptive algorithm A_r,n^ada with error satisfying ∥f - A_r,n^adaf∥_C ≤C_rT^r∥_L^∞n^-r for all n ≥ n₀ and n₀ independent of f. This result cannot be achieved for functions with more than just one singular point. However, the convergence rate n^-r can be recovered asymptotically by a nonadaptive algorithm A_r,n^-non that is a slightly modified A_r,n^non. Specifically, lim sup _n→∞ ∥f-A_r,n ^-nonf∥_C·n^r≤ C_rT ^r∥f^(r)∥_L^∞ for all r-smooth functions f with finitely many singular points.