On the dependence of the properties of the set of points of discontinuity

A. P. Petukhov

Sbornik. Mathematics, Tome 67 (1990) no. 2, pp. 427-447

Résumé

Let $c_\alpha(f)=\varliminf_{n\to\infty}nH_\alpha E_n(f)$, where $H_\alpha E_n(f)$ is the smallest deviation of a $2\pi$-periodic function $f$ from trigonometric polynomials of order $\leqslant n$ in the Hausdorff $\alpha$-metric. It is shown that for arbitrary $\alpha>0$ there exists a function $f_\alpha$ such that $c_\alpha(f_\alpha)=\pi/2\alpha$ and the set of points of discontinuity of $f_\alpha$ has Hausdorff dimension $1$. The concept of the $\sigma$-equiporosity coefficient $R(E)$ of a set $E$ is introduced, and a best possible lower estimate is obtained for the $\sigma$-equiporosity coefficient of the set $D(f)$ of points of discontinuity of a function $f$ in terms of the quantity $c_\alpha(f)$, $\pi/2\alpha\leqslant c_\alpha(f)\leqslant\pi/\alpha$: $$ R(D(f))\geqslant\frac{2(\pi-\alpha c_\alpha(f))}{3\pi-2\alpha c_\alpha(f)}. $$ Dolzhenko, Sevast'yanov, Petrushev, and Tashev proved earlier that the condition $c_\alpha(f)\pi/\alpha$ implies that $f$ is continuous almost everywhere, and $c_\alpha(f)\pi/2\alpha$ implies continuity at all points. Petrushev and Tashev constructed an example of a discontinuous function $f$ for which $c_\alpha(f)=\pi/2\alpha$, but, in contrast to the example mentioned above, $f$ had only one point of discontinuity on a period. Bibliography: 11 titles.