## Title data

Baier, Robert ; Dyn, N. ; Farkhi, E.:

**The Metric Average of 1D Compact Sets.**

*In:* Chui, Charles K. ; Schumaker, Larry L. ; Stöckler, Joachim
(ed.):
Approximation Theory X : selections of papers that were presented at the Tenth International Conference on Approximation Theory, held in St. Louis, Missouri, in March 2001. Volume 1. Abstract and classical analysis. -
Nashville
: Vanderbilt Univ. Press
,
2002
. - pp. 9-22
. - (Innovations in Applied Mathematics
)

ISBN 0-8265-1415-4

Review: |

## Related URLs

## Abstract in another language

We study properties of a binary operation between two compact sets depending on a weight in [0,1], termed metric average. The metric average is used in spline subdivision schemes for compact sets in |R^n, instead of the Minkowski convex combination of sets, to retain non-convexity, see N. Dyn, E. Farkhi, ``Spline subdivision schemes for compact sets with metric averages", Trends in Approximation Theory (2001).

Some properties of the metric average of sets in |R, like the cancellation property and the linear behavior of the Lebesgue measure of the metric average with respect to the weight, are proven. We present an algorithm for computing the metric average of two compact sets in |R, which are finite unions of intervals, as well as an algorithm for reconstructing one of the metric average's operands, given the second operand, the metric average and the weight.

## Further data

Item Type: | Article in a book |
---|---|

Refereed: | Yes |

Additional notes: | Contents:
1. Introduction 2. Definitions and Notation 3. Properties of the Metric Average 4. Algorithm for Computing the Metric Average 5. Cancellation Property 6. Proofs |

Keywords: | Metric average; Cancellation property; Finite union of intervals; Compact sets; Algorithm |

Subject classification: | Mathematics Subject Classification Code: 52A27 (65D18 65G30) |

Institutions of the University: | Faculties Faculties > Faculty of Mathematics, Physics und Computer Science Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics) Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics) > Chair Mathematics V (Applied Mathematics) - Univ.-Prof. Dr. Lars Grüne |

Result of work at the UBT: | Yes |

DDC Subjects: | 500 Science 500 Science > 510 Mathematics |

Date Deposited: | 01 Mar 2021 11:39 |

Last Modified: | 12 May 2021 05:47 |

URI: | https://eref.uni-bayreuth.de/id/eprint/63395 |