Functions with different strong and weak $F$-variations

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 ${\dim _s}(f) = 1/\alpha$.