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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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/

[1] Y. Katznelson, “Sur les ensembles de divergence des series trigonometriques”, Studia Math., 26 (1966), 301–304 | DOI | MR | Zbl

[2] Y. Katznelson, An Introduction to Harmonic Analysis, John Wiley Sons, New York, 1968 | MR | Zbl

[3] D. C. Harris, W. R. Wade, “Sets of divergence on the group $2^{\omega}$”, Trans. Amer. Math. Soc., 240 (1978), 385–392 | MR | Zbl

[4] Sh. V. Kheladze, “O raskhodimosti vsyudu ryadov Fure po ogranichennym sistemam Vilenkina”, Tr. Tbiliss. matem. in-ta, 58 (1978), 224–242 | MR | Zbl

[5] S. V. Bochkarev, “Abstraktnaya teorema Kolmogorova, prilozhenie k metricheskim prostranstvam i topologicheskim gruppam”, Matem. sb., 209:11 (2018), 32–59 | DOI | MR | Zbl

[6] T. P. Lukashenko, “Ob'edinenie mnozhestv raskhodimosti funktsionalnykh ryadov”, Matem. zametki, 31:4 (1982), 539–548 | MR | Zbl

[7] E. M. Stein, “On limits of seqences of operators”, Ann. of Math. (2), 183 (1961), 1327–1329 | MR

[8] A. N. Kolmogoroff, “Une série de Fourierp–Lebesgue divergente partout”, C. R. Acad. Sci. Paris, 74 (1926), 140–170

[9] V. I. Golubov, A. V. Efimov, V. A. Skvortsov, Ryady i preobrazovaniya Uolsha, Nauka, M., 1987 | MR

[10] F. Schipp, W. R. Wade, P. Simon, Walsh Series. An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990 | MR | Zbl

[11] F. Schipp, “Über die Grössenordnung der Partialsummen der Entwicklung integrierbarer Funktionen nach $W$-Systemen”, Acta Sci. Math. (Szeged), 28 (1967), 123–134 | MR | Zbl

[12] F. Schipp, “Über die Divergenz der Walsh-Fourierreihen”, Ann. Univ. Sci. Budapest. Eotvös Sect. Math., 12 (1969), 49–62 | MR | Zbl

[13] I. V. Polyakov, “Primery raskhodyaschikhsya ryadov Fure dlya shirokogo klassa perestavlennykh sistem Uolsha–Pelli”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2012, no. 6, 3–8 | MR

[14] W. S. Young, “On the a.e. convergence of Walsh–Kaczmarz–Fourier series”, Proc. Amer. Math. Soc., 44 (1974), 353–358 | DOI | MR | Zbl

[15] I. V. Polyakov, “Otsenki yadra Dirikhle i raskhodyaschiesya ryady Fure po sisteme Uolsha–Kachmazha”, Matem. zametki, 95:2 (2014), 257–270 | DOI | MR | Zbl

[16] S. V. Konyagin, “O raskhodimosti vsyudu podposledovatelnostei chastnykh summ trigonometricheskikh ryadov Fure”, Teoriya funktsii, Tr. IMM UrO RAN, 11, no. 2, 2005, 112–119 | MR | Zbl

[17] N. Ya. Vilenkin, “Ob odnom klasse polnykh ortonormalnykh sistem”, Izv. AN SSSR. Ser. matem., 11:4 (1947), 363–400 | MR | Zbl

[18] G. N. Agaev, N..Ya. Vilenkin, G. M. Dzhafarli, A. I. Rubinshtein, Multiplikativnye sistemy funktsii i garmonicheskii analiz na nul-mernykh gruppakh, Elm, Baku, 1981 | MR

[19] P. Simon, “On the divergence of Vilenkin-Fourier series”, Acta Math. Hungar., 41:3-4 (1983), 359–370 | DOI | MR | Zbl