Geodesic
On the Divergence Sets of Fourier Series in Systems of Characters of Compact Abelian Groups
Matematičeskie zametki, Tome 112 (2022) no. 1, pp. 95-105

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For a class of character systems of compact Abelian groups and for homogeneous Banach spaces $B$ satisfying some additional regularity conditions, we prove the following alternative: either the Fourier series of an arbitrary function in $B$ converges almost everywhere, or there exists a function in $B$ whose Fourier series diverges everywhere. We also prove that the classes of divergence sets of Fourier series in such function systems in the above-mentioned spaces are closed under at most countable unions and contain all sets of measure zero. As corollaries, we obtain some well-known and new results on everywhere divergent Fourier series in the trigonometric system as well as in the Walsh and Vilenkin systems and their rearrangements.
Keywords: Fourier series, compact Abelian group, character, divergence everywhere.
Mots-clés : divergence set 
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     author = {G. G. Oniani},
     title = {On the {Divergence} {Sets} of {Fourier} {Series} in {Systems} of {Characters} of {Compact} {Abelian} {Groups}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {95--105},
     publisher = {mathdoc},
     volume = {112},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_112_1_a9/}
}
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G. G. Oniani. On the Divergence Sets of Fourier Series in Systems of Characters of Compact Abelian Groups. Matematičeskie zametki, Tome 112 (2022) no. 1, pp. 95-105. http://geodesic.mathdoc.fr/item/MZM_2022_112_1_a9/