# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Functions with different strong and weak $F$-variationsHTML articles powered by AMS MathViewer

by Lane Yoder
Proc. Amer. Math. Soc. 56 (1976), 211-216 Request permission

## 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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