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 Mat. Sb., 1989, Volume 180, Number 7, Pages 969–988 (Mi msb1645)

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.

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English version:
Mathematics of the USSR-Sbornik, 1990, 67:2, 427–447

Bibliographic databases:

UDC: 517.51
MSC: Primary 26A15, 41A25, 42A10; Secondary 41A10

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

Citation in format AMSBIB
\Bibitem{Pet89} \by A.~P.~Petukhov \paper On the dependence of the properties of the set of points of discontinuity of a function on the degree of its polynomial Hausdorff approximations \jour Mat. Sb. \yr 1989 \vol 180 \issue 7 \pages 969--988 \mathnet{http://mi.mathnet.ru/msb1645} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1014624} \zmath{https://zbmath.org/?q=an:0754.42003} \transl \jour Math. USSR-Sb. \yr 1990 \vol 67 \issue 2 \pages 427--447 \crossref{https://doi.org/10.1070/SM1990v067n02ABEH002090} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1990EN23400007} 

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This publication is cited in the following articles:
1. A. I. Ermakov, “Best Hausdorff approximations by algebraic polynomials and porosity”, Math. Notes, 59:5 (1996), 498–506
2. Zajicek L., “On SIGMA-Porous Sets in Abstract Spaces”, Abstract Appl. Anal., 2005, no. 5, 509–534
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