### Abstract

An analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions. Interpolation and best approximation by polynomial splines, Hermite and finite element functions are examples of such methods. A direct theorem is proven for methods which are stable, quasi-linear and optimally accurate for sufficiently smooth functions. These assumptions are known to be satisfied in many cases of practical interest. Under a certain additional assumption, on the family of meshes, an inverse theorem is proven which shows that the direct theorem is sharp.

Original language | English (US) |
---|---|

Pages (from-to) | 327-338 |

Number of pages | 12 |

Journal | Numerische Mathematik |

Volume | 27 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1976 |

### Fingerprint

### Keywords

- AMS Subject Classifications: 41A15, 41A25, 41A40, 41A65, 46E35

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics
- Mathematics(all)

### Cite this

*Numerische Mathematik*,

*27*(3), 327-338. https://doi.org/10.1007/BF01396181

**On best error bounds for approximation by piecewise polynomial functions.** / Widlund, Olof.

Research output: Contribution to journal › Article

*Numerische Mathematik*, vol. 27, no. 3, pp. 327-338. https://doi.org/10.1007/BF01396181

}

TY - JOUR

T1 - On best error bounds for approximation by piecewise polynomial functions

AU - Widlund, Olof

PY - 1976/9

Y1 - 1976/9

N2 - An analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions. Interpolation and best approximation by polynomial splines, Hermite and finite element functions are examples of such methods. A direct theorem is proven for methods which are stable, quasi-linear and optimally accurate for sufficiently smooth functions. These assumptions are known to be satisfied in many cases of practical interest. Under a certain additional assumption, on the family of meshes, an inverse theorem is proven which shows that the direct theorem is sharp.

AB - An analog of the well-known Jackson-Bernstein-Zygmund theory on best approximation by trigonometric polynomials is developed for approximation methods which use piecewise polynomial functions. Interpolation and best approximation by polynomial splines, Hermite and finite element functions are examples of such methods. A direct theorem is proven for methods which are stable, quasi-linear and optimally accurate for sufficiently smooth functions. These assumptions are known to be satisfied in many cases of practical interest. Under a certain additional assumption, on the family of meshes, an inverse theorem is proven which shows that the direct theorem is sharp.

KW - AMS Subject Classifications: 41A15, 41A25, 41A40, 41A65, 46E35

UR - http://www.scopus.com/inward/record.url?scp=3042688198&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3042688198&partnerID=8YFLogxK

U2 - 10.1007/BF01396181

DO - 10.1007/BF01396181

M3 - Article

VL - 27

SP - 327

EP - 338

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 3

ER -