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Author Lerman, Gilad ♦ Whitehouse, J. Tyler
Source arXiv.org
Content type Text
File Format PDF
Date of Submission 2008-08-31
Language English
Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics
Subject Keyword Mathematics - Metric Geometry ♦ Mathematics - Functional Analysis ♦ 60D05 ♦ 49Q15 ♦ 42C99 ♦ math
Abstract This is the second of two papers wherein we estimate multiscale least squares approximations of certain measures by Menger-type curvatures. More specifically, we study an arbitrary d-regular measure on a real separable Hilbert space. The main result of the paper bounds the least squares error of approximation at any ball by an average of the discrete Menger-type curvature over certain simplices in in the ball. A consequent result bounds the Jones-type flatness by an integral of the discrete curvature over all simplices. The preceding paper provided the opposite inequalities. Furthermore, we demonstrate some other discrete curvatures for characterizing uniform rectifiability and additional continuous curvatures for characterizing special instances of the (p, q)-geometric property. We also show that a curvature suggested by Leger (Annals of Math, 149(3), p. 831-869, 1999) does not fit within our framework.
Description Reference: Constructive Approximation, 30(3): 325-360, 2009
Educational Use Research
Learning Resource Type Article
Page Count 32


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