On the Partial Sums of the Fourier Series of Functions of Bounded Variation
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 154 (2012) no. 3, pp. 121-128
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S. Banach [Sur la divergence des séries orthogonales. Studia Math., 1940, vol. 9, pp. 139–155] proved that for any function $f(x)\in L_2(I)$ $(I=[0,1]$, $f(x)\not\sim 0)$ there exists an orthonormal system (ONS) $(\varphi_n(x))$ such that $\varlimsup\limits_{n\to \infty} |S_n(f,x)|=+\infty$ almost everywhere on $I$, where $S_n(f,x)$ are the partial sums of the Fourier series of a function $f(x)$ with respect to the system $(\varphi_n(x))=\Phi$. This paper finds necessary and sufficient conditions which should be satisfied by ONS so that the partial sums of the Fourier series of functions with finite variation be uniformly bounded on $I$.
Keywords:
bounded variation, partial sums, subsystem.
@article{UZKU_2012_154_3_a11,
author = {L. D. Gogoladze and V. Sh. Tsagareishvili},
title = {On the {Partial} {Sums} of the {Fourier} {Series} of {Functions} {of~Bounded} {Variation}},
journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
pages = {121--128},
publisher = {mathdoc},
volume = {154},
number = {3},
year = {2012},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/UZKU_2012_154_3_a11/}
}
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L. D. Gogoladze; V. Sh. Tsagareishvili. On the Partial Sums of the Fourier Series of Functions of Bounded Variation. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 154 (2012) no. 3, pp. 121-128. http://geodesic.mathdoc.fr/item/UZKU_2012_154_3_a11/