On problems of the divergence of logarithmic means of Fourier series

Goginava, Ushangi; Mukhamedov, Farrukh

doi:10.1090/proc/17191

On problems of the divergence of logarithmic means of Fourier series
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by Ushangi Goginava and Farrukh Mukhamedov;

Proc. Amer. Math. Soc. 153 (2025), 2555-2564

DOI: https://doi.org/10.1090/proc/17191

Published electronically: March 24, 2025

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Abstract:

The present study provides an answer to certain long-standing divergence problems involving the logarithmic means of Fourier series in the context of general orthonormal systems.

References

Anas Ahmad Abu Joudeh and György Gát, Convergence of Cesáro means with varying parameters of Walsh-Fourier series, Miskolc Math. Notes 19 (2018), no. 1, 303–317. MR 3895589, DOI 10.18514/mmn.2018.2347
T. Akhobadze, On the convergence of generalized Cesàro means of trigonometric Fourier series. I, Acta Math. Hungar. 115 (2007), no. 1-2, 59–78. MR 2316617, DOI 10.1007/s10474-007-5214-7
T. Akhobadze and Sh. Zviadadze, A note on the generalized Cesáro means of trigonometric Fourier series, Izv. Nats. Akad. Nauk Armenii Mat. 54 (2019), no. 5, 3–10; English transl., J. Contemp. Math. Anal. 54 (2019), no. 5, 263–267. MR 4026067
S. V. Bočkarev, A Fourier series that diverges on a set of positive measure for an arbitrary bounded orthonormal system, Mat. Sb. (N.S.) 98(140) (1975), no. 3(11), 436–449, 496 (Russian). MR 390634
S. V. Bochkarev, Everywhere divergent Fourier series in the Walsh system and in multiplicative systems, Uspekhi Mat. Nauk 59 (2004), no. 1(355), 103–124 (Russian, with Russian summary); English transl., Russian Math. Surveys 59 (2004), no. 1, 103–124. MR 2069165, DOI 10.1070/RM2004v059n01ABEH000703
V. M. Bugadze, Divergence of Fourier-Walsh series of bounded functions on sets of measure zero, Mat. Sb. 185 (1994), no. 7, 119–127 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 82 (1995), no. 2, 365–372. MR 1300136, DOI 10.1070/SM1995v082n02ABEH003570
V. V. Buzdalin, Unboundedly diverging trigonometric Fourier series of continuous functions, Mat. Zametki 7 (1970), 7–18 (Russian). MR 262757
Francesco Di Plinio, Lacunary Fourier and Walsh-Fourier series near $L^1$, Collect. Math. 65 (2014), no. 2, 219–232. MR 3189278, DOI 10.1007/s13348-013-0094-3
Yen Q. Do and Michael T. Lacey, On the convergence of lacunacy Walsh-Fourier series, Bull. Lond. Math. Soc. 44 (2012), no. 2, 241–254. MR 2914604, DOI 10.1112/blms/bdr088
György Gát, Almost everywhere convergence of Fejér and logarithmic means of subsequences of partial sums of the Walsh-Fourier series of integrable functions, J. Approx. Theory 162 (2010), no. 4, 687–708. MR 2606641, DOI 10.1016/j.jat.2009.07.011
György Gát, Cesàro means of subsequences of partial sums of trigonometric Fourier series, Constr. Approx. 49 (2019), no. 1, 59–101. MR 3895764, DOI 10.1007/s00365-018-9438-2
György Gát, Almost everywhere divergence of Cesàro means of subsequences of partial sums of trigonometric Fourier series, Math. Ann. 389 (2024), no. 4, 4199–4231. MR 4768723, DOI 10.1007/s00208-023-02746-z
G. Gát and U. Goginava, Uniform and $L$-convergence of logarithmic means of Walsh-Fourier series, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 2, 497–506. MR 2214371, DOI 10.1007/s10114-005-0648-8
György Gát and Ushangi Goginava, On the divergence of Nörlund logarithmic means of Walsh-Fourier series, Acta Math. Sin. (Engl. Ser.) 25 (2009), no. 6, 903–916. MR 2511534, DOI 10.1007/s10114-009-7013-2
U. Goginava, On the divergence of Walsh-Fejér means of bounded functions on sets of measure zero, Acta Math. Hungar. 121 (2008), no. 4, 359–369. MR 2461440, DOI 10.1007/s10474-008-7219-2
Ushangi Goginava and Giorgi Oniani, On the divergence of subsequences of partial Walsh-Fourier sums, J. Math. Anal. Appl. 497 (2021), no. 2, Paper No. 124900, 13. MR 4197287, DOI 10.1016/j.jmaa.2020.124900
Richard P. Gosselin, On the divergence of Fourier series, Proc. Amer. Math. Soc. 9 (1958), 278–282. MR 96068, DOI 10.1090/S0002-9939-1958-0096068-0
G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 30620
J.-P. Kahane and Y. Katznelson, Sur les coefficients des séries de Fourier dont les sommes partielles sont positives sur un ensemble, Studia Math. 44 (1972), 555–562 (French). MR 340922, DOI 10.4064/sm-44-6-555-562
I. B. Kaplan, Cesàro means of variable order, Izv. Vysš. Učebn. Zaved. Matematika 1960 (1960), no. 5(18), 62–73 (Russian). MR 131696
G. A. Karagulyan, Divergence of general operators on sets of measure zero, Colloq. Math. 121 (2010), no. 1, 113–119. MR 2725706, DOI 10.4064/cm121-1-10
Jean-Pierre Kahane and Yitzhak Katznelson, Sur les ensembles de divergence des séries trigonométriques, Studia Math. 26 (1966), 305–306 (French). MR 199633, DOI 10.4064/sm-26-3-307-313
K. S. Kazaryan and P. L. Ul′yanov, On a problem of Alexits with respect to the divergence of orthogonal Fourier series, Dokl. Akad. Nauk SSSR 316 (1991), no. 3, 542–546 (Russian); English transl., Soviet Math. Dokl. 43 (1991), no. 1, 132–136. MR 1100154
A. Kolmogoroff, Une série de Fourier-Lebesgue divergente presque partout, Fund. Math. 4 (1923), no. 1, 324–328.
A. Kolmogoroff, Une série de Fourier-Lebesgue divergente partout, CR Acad. ScL Paris 183 (1926).
S. V. Konyagin, On a subsequence of Fourier-Walsh partial sums, Mat. Zametki 54 (1993), no. 4, 69–75, 158 (Russian, with Russian summary); English transl., Math. Notes 54 (1993), no. 3-4, 1026–1030 (1994). MR 1256607, DOI 10.1007/BF01210421
S. V. Konyagin, Divergence everywhere of subsequences of partial sums of trigonometric Fourier series, Proc. Steklov Inst. Math. Function Theory, suppl. 2 (2005), S167–S175. MR 2200228
Sergey V. Konyagin, Almost everywhere convergence and divergence of Fourier series, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1393–1403. MR 2275651
S. V. Konyagin, Almost everywhere divergence of lacunary subsequences of partial sums of Fourier series, Proc. Steklov Inst. Math. 273 (2011), no. suppl. 1, S99–S106. MR 3545867, DOI 10.1134/S0081543811050105
Victor Lie, On the pointwise convergence of the sequence of partial Fourier sums along lacunary subsequences, J. Funct. Anal. 263 (2012), no. 11, 3391–3411. MR 2984070, DOI 10.1016/j.jfa.2012.08.013
F. Móricz and A. H. Siddiqi, Approximation by Nörlund means of Walsh-Fourier series, J. Approx. Theory 70 (1992), no. 3, 375–389. MR 1178380, DOI 10.1016/0021-9045(92)90067-X
S. Stechkin, On convergence and divergence of trigonometric series. Uspekhi Mat. Nauk 6 (1951), no. 2, 148–149.
E. M. Stein, On limits of seqences of operators, Ann. of Math. (2) 74 (1961), 140–170. MR 125392, DOI 10.2307/1970308
Shakro Tetunashvili, On divergence of Fourier series by some methods of summability, J. Funct. Spaces Appl. , posted on (2012), Art. ID 542607, 9. MR 2873712, DOI 10.1155/2012/542607
V. Totik, On the divergence of Fourier series, Publ. Math. Debrecen 29 (1982), no. 3-4, 251–264. MR 678901, DOI 10.5486/pmd.1982.29.3-4.05
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 236587

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On problems of the divergence of logarithmic means of Fourier series
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Abstract: