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1 paper
Convergence of Trigonometric Fourier Series of Functions with a Constraint on the Fractality of Their Graphs
M. L. Gridnev Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
Abstract:
For a function
$f$ continuous on a closed interval, its modulus of fractality
$\nu(f,\varepsilon)$ is defined as the function that maps any
$\varepsilon>0$ to the smallest number of squares of size
$\varepsilon$ that cover the graph of
$f$. The following condition for the uniform convergence of the Fourier series of
$f$ is obtained in terms of the modulus of fractality and the modulus of continuity
$\omega(f,\delta)$: if
$$
\omega (f,\pi/n) \ln\bigg(\frac{\nu(f,\pi/n)}{n}\bigg) \longrightarrow 0\ \ \
as \ n\longrightarrow+\infty,
$$
then the Fourier series of
$f$ converges uniformly. This condition refines the known Dini–Lipschitz test. In addition, for the growth order of the partial sums
$S_n(f,x)$ of a continuous function
$f$, we derive an estimate that is uniform in
$x\in[0,2\pi]$:
$$
S_n(f,x)=o\bigg( \ln \bigg(\frac{\nu (f,\pi / n)}{n}\bigg)\bigg).
$$
The optimality of this estimate is shown.
Keywords:
trigonometric Fourier series, uniform convergence, fractal dimension.
UDC:
517.518.45
MSC: 42A20 Received: 31.08.2018
Revised: 28.10.2018
Accepted: 05.11.2018
DOI:
10.21538/0134-4889-2018-24-4-104-109