Abstract
In this paper, we study the a.e. exponential strong summability problem for the rectangular partial sums of double trigonometric Fourier series of functions in L logL.
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L. Fejér, “Untersuchungen über Fouriersche Reihen,” Math. Ann. 58, 51–69 (1904).
H. Lebesgue, “Recherches sur la sommabilitéforte des series de Fourier,” Math. Ann. 61, 251–280 (1905).
G. H. Hardy and J. E. Littlewood, “Sur la séries de Fourier d’une fonction à carre sommable,” Comptes Rendus (Paris) 156, 1307–1309 (1913).
J. Marcinkiewicz, “Sur unemethode remarquable de sommation des séries doublées de Fourier,” Ann. Scuola Norm. Sup. Pisa 8, 149–160 (1939).
A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 1959).
K. I. Oskolkov, “On strong summability of Fourier series,” in Trudy Mat. Inst. Steklov Vol. 172: Studies in the Theory of Functions of Several Real Variables and Approximation of Functions, Collection of papers (Nauka, Moscow, 1985), pp. 280–290 [in Russian] [Proc. Steklov Inst.Math. 172, 303–314 (1987)].
V. A. Rodin, “The space BMO and strong means of Fourier series,” Anal.Math. 16 (4), 291–302 (1990).
G. A. Karagulyan, “On the divergence of strong F-means of Fourier series,” J. Contemp.Math. Anal. 26 (2), 66–69 (1991).
G. A. Karagulyan, “Everywhere divergent F-means of Fourier series,” Mat. Zametki 80 (1), 50–59 (2006) [Math. Notes 80 (1), 47–56 (2006)].
B. Jessen, J. Marcinkiewicz, and A. Zygmund, “Note on the differentiability of multiple integrals,” Fund. Math. 25, 217–234 (1935).
L. D. Gogoladze, “On strong summability almost everywhere,” Mat. Sb. 135 (177) (2), 158–168 (1988) [Math. USSR-Sb. 63 (1), 153–164 (1989)].
F. Schipp, “Über die starke Summation von Walsh–Fourier Reihen,” Acta Sci. Math. (Szeged) 30, 77–87 (1969).
F. Schipp, “On strong approximation ofWalsh–Fourier series,” Magyar Tud. Akad.Mat. Fiz.Oszt. Közl. 19, 101–111 (1969).
F. Schipp and N. X. Ky, “On strong summability of polynomial expansions,” Anal. Math. 12 (2), 115–127 (1986).
L. Leindler, “On the strong approximation of Fourier series,” Acta Sci.Math. (Szeged) 38, 317–324 (1976).
L. Leindler, “Strong approximation and classes of functions,” Mitt.Math. Sem. Giessen 132, 29–38 (1978).
L. Leindler, Strong Approximation by Fourier Series (Akademiai Kiado, Budapest, 1985).
L. Leindler, “Über die Approximation im starken Sinne,” Acta Math. Acad. Sci. Hungar. 16, 255–262 (1965).
V. Totik, “On the strong approximation of Fourier series,” ActaMath. Acad. Sci.Hungar. 35 (1-2), 151–172 (1980).
V. Totik, “On the strong approximation of Fourier series,” ActaMath. Acad. Sci.Hungar. 35 (1-2), 151–172 (1980).
V. Totik, “On the generalization of Fejér’s summation theorem,” in Functions, Series, Operators, Colloq. Math. Soc. János Bolyai (North Holland, Amsterdam–Oxford–New York, 1983), Vol. 35, pp. 1195–1199.
V. Totik, “Notes on Fourier series: strong approximation,” J. Approx. Theory 43, 105–111 (1985).
U. Goginava and L. Gogoladze, “Strong approximation by Marcinkiewicz means of double Walsh–Fourier series,” Constr. Approx. 35 (1), 1–19 (2012).
U. Goginava and L. Gogoladze, “Strong approximation of doubleWalsh–Fourier series,” Studia Sci. Math. Hungar. 49 (2), 170–188 (2012).
U. Goginava, L. Gogoladze, and G. Karagulyan, “BMO-estimation and almost everywhere exponential summability of quadratic partial sums of double Fourier series,” Constr. Approx. 40 (1), 105–120 (2014).
G. Gát, U. Goginava, and G. Karagulyan, “Almost everywhere strong summability of Marcinkiewicz means of doubleWalsh–Fourier series,” Anal.Math. 40 (4), 243–266 (2014).
G. Gát, U. Goginava, and G. Karagulyan, “On everywhere divergence of the strong F-means of Walsh–Fourier series,” J. Math. Anal. Appl. 421 (1), 206–214 (2015).
F. Weisz, “Strong summability of more-dimensional Ciesielski–Fourier series,” East J. Approx. 10 (3), 333–354 (2004).
F. Weisz, “Lebesgue points of double Fourier series and strong summability,” J. Math. Anal. Appl. 432 (1), 441–462 (2015).
F. Weisz, “Lebesgue points of two-dimensional Fourier transforms and strong summability,” J. Fourier Anal. Appl. 21 (4), 885–914 (2015).
F. Weisz, “Strong summability of Fourier transforms at Lebesgue points and Wiener amalgam spaces,” J. Funct. Spaces, No. 420750 (2015).
O. D. Gabisoniya, “Points of strong summability of Fourier series,” Mat. Zametki 14 (5), 615–626 (1973) [Math. Notes 14 (5), 913–918 (1973)].
F. Schipp, “On the strong summability of Walsh series,” Publ.Math. Debrecen 52 (3-4), 611–633 (1998).
M. A. Krasnosel’skii and Ya. B. Rutitskii, Convex Functions and Orlicz Spaces (Fizmatgiz, Moscow, 1958) [in Russian].
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Original Russian Text © U. Goginava, G. Karagulyan, 2018, published in Matematicheskie Zametki, 2018, Vol. 104, No. 5, pp. 667–679.
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Goginava, U., Karagulyan, G. On Exponential Summability of Rectangular Partial Sums of Double Trigonometric Fourier Series. Math Notes 104, 655–665 (2018). https://doi.org/10.1134/S0001434618110056
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DOI: https://doi.org/10.1134/S0001434618110056