## Abstract

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}^{ada}f∥_{C} ≤C_{r}T^{r}∥_{L}^{∞}n^{-r} for all n ≥ n_{0} and n_{0} 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} ^{-non}f∥_{C}·n^{r}≤ C_{r}T ^{r}∥f^{(r)}∥_{L}^{∞} for all r-smooth functions f with finitely many singular points.

Original language | English |
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Pages (from-to) | 762-785 |

Number of pages | 24 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 47 |

Issue number | 1 |

DOIs | |

State | Published - 2008 |

## Keywords

- Adaptive algorithms
- Function approximation
- Singular functions

## ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics