Functions with different strong and weak $F$variations
Author:
Lane Yoder
Journal:
Proc. Amer. Math. Soc. 56 (1976), 211216
MSC:
Primary 26A45
DOI:
https://doi.org/10.1090/S00029939197603993813
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 $\sup \Sigma \Phi (f({x_i})  f({x_{i  1}})) < \infty$ for the partitions of $[a,b]$. Since $\inf \Sigma \Phi (f({x_i})  f({x_{i  1}})) = 0$ 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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© Copyright 1976
American Mathematical Society