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The Metric Average of 1D Compact Sets

DOI zum Zitieren der Version auf EPub Bayreuth: https://doi.org/10.15495/EPub_UBT_00005491
URN to cite this document: urn:nbn:de:bvb:703-epub-5491-8

Title data

Baier, Robert ; Dyn, N. ; Farkhi, Elza:
The Metric Average of 1D Compact Sets.
Bayreuth , 2002

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Abstract

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: Preprint, postprint
Additional notes (visible to public): Erscheint in: Chui, Charles K. ; Schumaker, Larry L. ; Stöckler, Joachim (Hrsg.): 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 . - S. 9-22
Keywords: Metric average; Cancellation property; Finite union of intervals; Compact sets; Algorithm
DDC Subjects: 500 Science
500 Science > 510 Mathematics
Institutions of the University: 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
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)
Language: English
Originates at UBT: Yes
URN: urn:nbn:de:bvb:703-epub-5491-8
Date Deposited: 12 May 2021 05:47
Last Modified: 21 Jun 2021 08:59
URI: https://epub.uni-bayreuth.de/id/eprint/5491

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