Convergence of Trigonometric Fourier Series of Functions with a Constraint on the Fractality of Their Graphs

M. L. Gridnev

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 4, pp. 104-109

Résumé

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.

Détail

Keywords: trigonometric Fourier series
Mots-clés : uniform convergence, fractal dimension.