This article is cited 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.
UDC:
517.51
MSC: Primary
26A15,
41A25,
42A10; Secondary
41A10 Received: 28.01.1988
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