Uniform approximation of piecewise r-smooth and globally continuous functions

Leszek Plaskota, Grzegorz W. Wasilkowski

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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 Ar,nnon 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 Ar,n non by using (r + 1) ⌊(r - 1)/2⌋ extra function evaluations to obtain an adaptive algorithm Ar,nada with error satisfying ∥f - Ar,nadaf∥C ≤CrTrLn-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 Ar,n-non that is a slightly modified Ar,nnon. Specifically, lim sup n→∞ ∥f-Ar,n -nonf∥C·nr≤ CrT r∥f(r)L for all r-smooth functions f with finitely many singular points.

Original languageEnglish
Pages (from-to)762-785
Number of pages24
JournalSIAM Journal on Numerical Analysis
Issue number1
StatePublished - 2008


  • Adaptive algorithms
  • Function approximation
  • Singular functions

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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