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Functions with different strong and weak $ F$-variations


Author: Lane Yoder
Journal: Proc. Amer. Math. Soc. 56 (1976), 211-216
MSC: Primary 26A45
DOI: https://doi.org/10.1090/S0002-9939-1976-0399381-3
MathSciNet review: 0399381
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Abstract: This paper shows by example how different the strong $ \Phi $-variation can be from the weak $ \Phi $-variation. Let $ \Phi $ be a convex function on $ [0,\infty )$ with $ \Phi (0) = 0$. A continuous function $ f$ on $ [a,b]$ is of bounded strong $ \Phi $-variation if $ \sup \Sigma \Phi (\vert f({x_i}) - f({x_{i - 1}})\vert) < \infty $ for the partitions of $ [a,b]$. Since $ \inf \Sigma \Phi (\vert f({x_i}) - f({x_{i - 1}})\vert) = 0$ if $ {\lim _{x \to 0}}{x^{ - 1}}\Phi (x) = 0$, the weak $ \Phi $-variation is defined as $ \inf \Sigma \Phi (\omega (f;{x_{i - 1}},{x_i}))$, where $ \omega (f;c,d)$ is the oscillation of $ f$ on $ [c,d]$. Of special interest is the case $ \Phi (x) = {x^p},p \geqslant 1$, in terms of which strong and weak variation dimensions are defined. They are denoted by $ {\dim _{\text{s}}}(f)$ and $ {\dim _{\text{w}}}(f)$, respectively. By a lemma of Goffman and Loughlin, the Hausdorff dimension $ d$ of the graph of $ f$ provides a lower bound for $ {\dim _w}(f):1/(2 - d) \leqslant {\dim _w}(f)$. A Lipschitz condition of order a provides an upper bound for $ {\dim _s}(f):{\dim _s}(f) \leqslant 1/\alpha $. Besicovitch and Ursell showed that $ 1 \leqslant d \leqslant 2 - \alpha $ and gave examples to show that $ d$ can take on any value in this interval. It turns out that these examples provide the extreme cases for variation dimensions; i.e., $ {\dim _w}(f) = 1/(2 - d)$ and $ {\dim _s}(f) = 1/\alpha $.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0399381-3
Article copyright: © Copyright 1976 American Mathematical Society

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