Geodesic
Convergence of Trigonometric Fourier Series of Functions with a Constraint on the Fractality of Their Graphs
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 4, pp. 104-109
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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
Mots-clés : uniform convergence, fractal dimension. 
@article{TIMM_2018_24_4_a7,
     author = {M. L. Gridnev},
     title = {Convergence of {Trigonometric} {Fourier} {Series} of {Functions} with a {Constraint} on the {Fractality} of {Their} {Graphs}},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {104--109},
     year = {2018},
     volume = {24},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2018_24_4_a7/}
}
TY  - JOUR
AU  - M. L. Gridnev
TI  - Convergence of Trigonometric Fourier Series of Functions with a Constraint on the Fractality of Their Graphs
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2018
SP  - 104
EP  - 109
VL  - 24
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TIMM_2018_24_4_a7/
LA  - ru
ID  - TIMM_2018_24_4_a7
ER  - 
%0 Journal Article
%A M. L. Gridnev
%T Convergence of Trigonometric Fourier Series of Functions with a Constraint on the Fractality of Their Graphs
%J Trudy Instituta matematiki i mehaniki
%D 2018
%P 104-109
%V 24
%N 4
%U http://geodesic.mathdoc.fr/item/TIMM_2018_24_4_a7/
%G ru
%F TIMM_2018_24_4_a7
M. L. Gridnev. Convergence of Trigonometric Fourier Series of Functions with a Constraint on the Fractality of Their Graphs. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 4, pp. 104-109. http://geodesic.mathdoc.fr/item/TIMM_2018_24_4_a7/

[1] Gridnev M.L., “O klassakh funktsii s ogranicheniem na fraktalnost ikh grafika”, Proc. of the 48th Internat. Youth School-Conf. “Modern Problems in Mathematics and its Applications”, CEUR Workshop Proceedings, 1894, Yekaterinburg, 2017, 167–173 URL: 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., 3:2 (2017), 46–50 | DOI | MR

[3] Bari N.K., Trigonometricheskie ryady, GIMFL, M., 1961, 937 pp.