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Quantifying curvelike structures of measures by using L 2 Jones quantities
Author(s) -
Lerman Gilad
Publication year - 2003
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10096
Subject(s) - measure (data warehouse) , mathematics , function (biology) , real line , line (geometry) , mathematical analysis , statistics , geometry , computer science , data mining , evolutionary biology , biology
We study the curvelike structure of special measures on ℝ n in a multiscale fashion. More precisely, we consider the existence and construction of a sufficiently short curve with a sufficiently large measure. Our main tool is an L 2 variant of Jones' β numbers, which measure the scaled deviations of the given measure from a best approximating line at different scales and locations. The Jones function is formed by adding the squares of the L 2 Jones numbers at different scales and the same location. Using a special L 2 Jones function, we construct a sufficiently short curve with a sufficiently large measure. The length and measure estimates of the underlying curve are expressed in terms of the size of this Jones function. © 2003 Wiley Periodicals, Inc.