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This article is cited in 2 scientific papers (total in 2 papers)
On the dependence of the properties of the set of points of discontinuity
of a function on the degree of its polynomial Hausdorff approximations
A. P. Petukhov
Abstract:
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.
Received: 28.01.1988
Citation:
A. P. Petukhov, “On the dependence of the properties of the set of points of discontinuity
of a function on the degree of its polynomial Hausdorff approximations”, Mat. Sb., 180:7 (1989), 969–988; Math. USSR-Sb., 67:2 (1990), 427–447
Linking options:
https://www.mathnet.ru/eng/sm1645https://doi.org/10.1070/SM1990v067n02ABEH002090 https://www.mathnet.ru/eng/sm/v180/i7/p969
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Abstract page: | 305 | Russian version PDF: | 78 | English version PDF: | 2 | References: | 39 | First page: | 1 |
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