## Abstract

We study the ω -weighted L^{p} 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∥L^{1/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 F̌^{1}_{r} 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 language | English |
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Pages (from-to) | 1470-1493 |

Number of pages | 24 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 51 |

Issue number | 3 |

DOIs | |

State | Published - 2013 |

## Keywords

- Adaptive algorithms
- Sampling
- Singularities
- Weighted approximation

## ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics