A rough differentiable function

Kirchheim, Bernd; Müller, Paul

doi:10.1090/S0002-9939-08-09322-2

A rough differentiable function
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by Bernd Kirchheim and Paul F.X. Müller

Proc. Amer. Math. Soc. 136 (2008), 3875-3881

DOI: https://doi.org/10.1090/S0002-9939-08-09322-2

Published electronically: June 26, 2008

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Abstract:

A real-valued continuously differentiable function $f$ on the unit interval is constructed such that \[ \sum _{k=1}^\infty \beta _f (x, 2^{-k} ) = \infty \] 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

Christopher J. Bishop and Peter W. Jones, Harmonic measure, $L^2$ estimates and the Schwarzian derivative, J. Anal. Math. 62 (1994), 77–113. MR 1269200, DOI 10.1007/BF02835949
J. Bourgain, On the radial variation of bounded analytic functions on the disc, Duke Math. J. 69 (1993), no. 3, 671–682. MR 1208816, DOI 10.1215/S0012-7094-93-06928-1
Zh. Burgeĭn, Boundedness of variation of convolution of measures, Mat. Zametki 54 (1993), no. 4, 24–33, 158 (Russian, with Russian summary); English transl., Math. Notes 54 (1993), no. 3-4, 995–1001 (1994). MR 1256604, DOI 10.1007/BF01210418
Casper Goffman, Real functions, Rinehart & Co., Inc., New York, N. Y., 1953. MR 0054006
Peter W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), no. 1, 1–15. MR 1069238, DOI 10.1007/BF01233418
P.W. Jones, private communication.
Ole Jørsboe, Leif Mejlbro, and Flemming Topsøe, Some Vitali theorems for Lebesgue measure, Math. Scand. 48 (1981), no. 2, 259–285. MR 631341, DOI 10.7146/math.scand.a-11917