An adaptive algorithm for weighted approximation of singular functions over ℝ

Leszek Plaskota, Grzegorz W. Wasilkowski, Yaxi Zhao

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

We study the ω -weighted Lp approximation (1 ≤ p ≤ ∞ ) of piecewise r-smooth functions f : ℝ → ℝ . Approximations Anf are based on n values of f at points that can be chosen adaptively. Assuming that the weight Σ is Riemann integrable on any compact interval and asymptotically decreasing, a necessary condition for the error of approximation to be of order n-r is that ∥ Σ∥L1/γ < ∞ , where γ = r+1/p. For the class Wγ of globally γ-smooth functions, this condition is also sufficient. Indeed, we show a nonadaptive algorithm P* n with the worst case error supf(eqution presented) n-rSuch worst case result does not hold in general for the class of piecewise r-smooth functions. However, if p < ∞ and the class is restricted to F̌1r of functions with at most one singularity and uniformly bounded singularity jumps, then an adaptive algorithm A *n can be constructed whose worst case error satisfies sup f (eqution presented) A modification of A.n gives an adaptive algorithm A*n such that the error (eqution presented) max (eqution) is of order n-r for any function f with finitely many singular points and with no restrictions on the jumps. For those results to hold, the use of adaption and p < ∞ is necessary. Yet similar results can be obtained if the error is measured in the weighted Skorohod metric instead of the weighted L∞ norm.

Original languageEnglish
Pages (from-to)1470-1493
Number of pages24
JournalSIAM Journal on Numerical Analysis
Volume51
Issue number3
DOIs
StatePublished - 2013

Keywords

  • Adaptive algorithms
  • Sampling
  • Singularities
  • Weighted approximation

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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