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A rough differentiable function


Authors: Bernd Kirchheim and Paul F.X. Müller
Journal: Proc. Amer. Math. Soc. 136 (2008), 3875-3881
MSC (2000): Primary 26A16, 30D55, 26A24, 30C99
DOI: https://doi.org/10.1090/S0002-9939-08-09322-2
Published electronically: June 26, 2008
MathSciNet review: 2425727
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Abstract | References | Similar Articles | Additional Information

Abstract: A real-valued continuously differentiable function $ f$ on the unit interval is constructed such that

$\displaystyle \sum_{k=1}^\infty \beta_f (x, 2^{-k} ) = \infty $

holds for every $ x \in [0,1].$ Here $ \beta_f (x, 2^{-k} )$ measures the distance of $ f$ to the best approximating linear function at scale $ 2^{-k}$ around $ x$.


References [Enhancements On Off] (What's this?)

  • [B-J] C. Bishop, P.W. Jones, Harmonic measure, $ L^2$-estimates, and the Schwarzian derivative, J. Anal. Math. 62 (1994), 77-113. MR 1269200 (95f:30034)
  • [B-1] J. Bourgain, On the radial variation of bounded analytic functions on the disk, Duke Math. J. 69 (1993), 671-682. MR 1208816 (94d:30061)
  • [B-2] J. Bourgain, Bounded variation of convolution of measures, Math. Zametki 54/4 (1993), 24-33. MR 1256604 (95i:30031)
  • [G] C. Goffman, Real Functions, 1953, Prindle, Weber & Schmidt, Inc., Boston. MR 0054006 (14:855e)
  • [J] P.W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), 1-15. MR 1069238 (91i:26016)
  • [J1] P.W. Jones, private communication.
  • [J-M-T] O. Jorsboe, L. Melbroe, F. Topsoe, Some Vitali theorems for Lebesgue measure, Math. Scand. 48 (1981), 259-285. MR 631341 (84h:28008)

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Additional Information

Bernd Kirchheim
Affiliation: Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, United Kingdom

Paul F.X. Müller
Affiliation: Institut für Analysis und Numerik, J. Kepler Universität Linz, A-4040 Linz, Austria
Email: pfxm@bayou.uni-linz.ac.at

DOI: https://doi.org/10.1090/S0002-9939-08-09322-2
Received by editor(s): May 17, 2002
Received by editor(s) in revised form: July 18, 2007
Published electronically: June 26, 2008
Communicated by: David Preiss
Article copyright: © Copyright 2008 by the authors

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