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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Representation of a crinkled arc


Author: Richard A. Vitale
Journal: Proc. Amer. Math. Soc. 52 (1975), 303-304
MSC: Primary 46C10
DOI: https://doi.org/10.1090/S0002-9939-1975-0388056-1
MathSciNet review: 0388056
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Abstract: Johnson [A crinkled arc, Proc. Amer. Math. Soc. 25 (1970), 375-376] has shown that under suitable normalizations all crinkled arcs are unitarily equivalent. Using this result, we find a general series expansion for a crinkled arc: \[ f(t) = \sqrt {2} \sum _{n=1}^\infty x_n \tfrac {\sin (n - 1/2)\pi t}{(n - 1/2)\pi } , \] where $\{ {x_n}\}$ is an orthonormal set.


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Keywords: Crinkled arc, Brownian motion, Karhunen-Loève expansion
Article copyright: © Copyright 1975 American Mathematical Society