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

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

REFERENCES

1.   Gridnev M. L. On classes of functions with a restriction on the fractality of their graphs. In: A. A. Makhnev, S. F. Pravdin (eds.): Proc. of the 48th Internat. Youth School-Conf. “Modern Problems in Mathematics and its Applications”, Yekaterinburg, 2017, vol. 1894, pp. 167–173 (in Russian). Published at http://ceur-ws.org/Vol-1894/appr5.pdf .

2.   Gridnev M. L. Divergence of Fourier series of continuous functions with restriction on the fractality of their graphs. Ural Math. J., 2017, vol. 3, no. 2, pp. 46–50. doi: 10.15826/umj.2017.2.007 .

3.   Bary N.K. A treatise on trigonometric series, vol. I; II. Oxford; N Y: Pergamon Press, 1964, 553 p.; 508 p. doi: 10.1002/zamm.19650450531 . Original Russian text published in Bari N.K. Trigonometricheskie ryady, Moscow: GIMFL Publ., 1961, 937 p.