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On regularized distance and related functions

Published online by Cambridge University Press:  14 November 2011

L. E. Fraenkel
L. E. Fraenkel
Affiliation:
Mathematics Division, University of Sussex
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Synopsis

Let F be any closed subset of ℝN. Stein's regularized distance is a smooth (C∞) function, defined on the complement cF, that approximates the distance from F of any point x ∈ cF in the manner shown by the inequalities (*) in the Introduction below. In this paper we use a method different from Stein's to construct a one-parameter family of smooth approximations to any positive Lipschitz continuous function, with the effect that the constants in (*) can be made arbitrarily close to 1. It is shown that partial derivatives of order two or more, while necessarily unbounded, are best possible in order of magnitude.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 83 , Issue 1-2 , 1979 , pp. 115 - 122
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

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