M.L. Gridnev. Convergence of trigonometric Fourier series of functions with a constraint on the fractality of their graphs ... P. 104-109

Full text (in Russian)

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 \ \mbox{ 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

Received August 31, 2018

Revised October 28, 2018

Accepted November 05, 2018

Funding Agency: This work was supported by the Russian Science Foundation (project no. 14-11-00702).

Maksim Leonidovich Gridnev, Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia, e-mail: coraxcoraxg@gmail.com


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