Functions with different strong and weak -variations

Author:
Lane Yoder

Journal:
Proc. Amer. Math. Soc. **56** (1976), 211-216

MSC:
Primary 26A45

MathSciNet review:
0399381

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Abstract: This paper shows by example how different the strong -variation can be from the weak -variation. Let be a convex function on with . A continuous function on is of bounded strong -variation if for the partitions of . Since if , the weak -variation is defined as , where is the oscillation of on . Of special interest is the case , in terms of which strong and weak variation dimensions are defined. They are denoted by and , respectively. By a lemma of Goffman and Loughlin, the Hausdorff dimension of the graph of provides a lower bound for . A Lipschitz condition of order a provides an upper bound for . Besicovitch and Ursell showed that and gave examples to show that can take on any value in this interval. It turns out that these examples provide the extreme cases for variation dimensions; i.e., and .

**[1]**A. S. Besicovitch and H. D. Ursell,*Sets of fractional dimension*. V:*On dimensional members of some continuous curves*, J. London Math. Soc.**12**(1937), 18-25.**[2]**Casper Goffman and John J. Loughlin,*Strong and weak Φ-variation of Brownian motion*, Indiana Univ. Math. J.**22**(1972/73), 135–138. MR**0296227****[3]**S. J. Taylor,*Exact asymptotic estimates of Brownian path variation*, Duke Math. J.**39**(1972), 219–241. MR**0295434****[4]**Lane Yoder,*The Hausdorff dimensions of the graph and range of 𝑁-parameter Brownian motion in 𝑑-space*, Ann. Probability**3**(1975), 169–171. MR**0359033****[5]**Lane Yoder,*Variation of multiparameter Brownian motion*, Proc. Amer. Math. Soc.**46**(1974), 302–309. MR**0418260**, 10.1090/S0002-9939-1974-0418260-4

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DOI:
https://doi.org/10.1090/S0002-9939-1976-0399381-3

Article copyright:
© Copyright 1976
American Mathematical Society