## Abstract

We study the error in approximating functions with a bounded (r + α)th derivative in an L_{p}-norm. Here r is a nonnegative integer, α ε{lunate} [0, 1), and f{hook}^{(r + α)} is the classical fractional derivative, i.e., f{hook}^{(r + α)}(y) = ∝_{0}^{1}, ^{α} d(f{hook}^{(r)}(t)). We prove that, for any such function f{hook}, there exists a piecewise-polynomial of degree s that interpolates f{hook} at n equally spaced points and that approximates f{hook} with an error (in sup-norm) ∥f{hook}^{(r + α)}∥_{p} O(n^{-(r+α- 1 p}). We also prove that no algorithm based on n function and/or derivative values of f{hook} has the error equal ∥f{hook}^{(r + α)}∥_{p} O(n^{-(r+α- 1 p}) for any f{hook}. This implies the optimality of piecewise-polynomial interpolation. These two results generalize well-known results on approximating functions with bounded rth derivative (α = 0). We stress that the piecewise-polynomial approximation does not depend on α nor on p. It does not depend on the exact value of r as well; what matters is an upper bound s on r, s ≥ r. Hence, even without knowing the actual regularity (r, α, and p) of f{hook}, we can approximate the function f{hook} with an error equal (modulo a constant) to the minimal worst case error when the regularity were known.

Original language | English |
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Pages (from-to) | 372-380 |

Number of pages | 9 |

Journal | Journal of Approximation Theory |

Volume | 62 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1990 |

## ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- General Mathematics
- Applied Mathematics

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