ON \(\Lambda\)-CONVERGENCE ALMOST EVERYWHERE OF MULTIPLE TRIGONOMETRIC FOURIER SERIES
Abstract
We consider one type of convergence of multiple trigonometric Fourier series intermediate between the convergence over cubes and the \(\lambda \)-convergence for \(\lambda >1\). The well-known result on the almost everywhere convergence over cubes of Fourier series of functions from the class \( L (\ln ^ + L) ^ d \ln ^ + \ln ^ + \ln ^ + L ([0,2 \pi)^d ) \) has been generalized to the case of the \( \Lambda \)-convergence for some sequences \(\Lambda\).
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Luzin N.N. Integral and trigonometric series. Moscow – Leningrad: GITTL. 1951. 550 p. [in Russian]
Kolmogoroff A. Une série de Fourier – Lebesgue divergente preque partout // Fund. math., 1923.Vol. 4. P. 324–328.
Kolmogoroff A. Une série de Fourier – Lebesgue divergente partout // C. r. Acad. sci. Paris, 1926. Vol. 183. P. 1327–1329.
Carleson L. On convergence and growth of partial sums of Fourier series // Acta math., 1966. Vol. 116, no. 1–2. P. 135–157. DOI:10.1007/BF02392815
Hunt R.A. On the convergence of Fourier series // Orthogonal expansions and their continuous analogues. SIU Press, Carbondale, Illinois, 1968. P. 235–255.
Sjölin P. An inequality of Paley and convergence a.e. of Walsh-Fourier series // Arkiv för mat., 1969. Vol. 7. P. 551–570. DOI: 10.1007/BF02590894
Antonov N.Yu. Convergence of Fourier series // East Journal on Approximations, 1996. Vol. 2, no. 2. P. 187–196.
Konyagin S.V. On everywhere divergence of trigonometric Fourier series// Sb. Math., 2000. Vol. 191, no. 1. P. 97–120. DOI: 10.1070/SM2000v191n01ABEH000449
Tevzadze N.R. The convergence of the double Fourier series of a square integrable function // Soobshch. AN GSSR, 1970. Vol. 58, no. 2. P. 277–279. [in Russian]
Fefferman C. On the convergence of multiple Fourier series // Bull. Amer. Math. Soc., 1971. Vol. 77, no. 5. P. 744–745. DOI: 10.1090/S0002-9904-1971-12793-3
Sjölin P. Convergence almost everywhere of ertain singular integrals and multiple Fourier series // Arkiv för mat., 1971. Vol. 9, no. 1. P. 65–90. DOI: 10.1007/BF02383638
Antonov N.Yu. Almost everywhere convergence over cubes of multiple trigonometric Fourier series // Izv. Math., 2004. Vol. 68, no. 2, P. 223–241. DOI: 10.1070/IM2004v068n02ABEH000472
Antonov N.Yu. On the almost everywhere convergence of sequences of multiple rectangular Fourier sums // Proc. Steklov Inst. Math., 2009. Vol. 264, Suppl. 1. P. S1–S18. DOI: 10.1134/S0081543809050010
Konyagin S.V. On divergence of trigonometric Fourier series over cubes // Acta Sci. Math. (Szeged), 1995. Vol. 61. P. 305–329.
Fefferman C. On the divergence of multiple Fourier series // Bull. Amer. Math. Soc., 1971. Vol. 77, no. 2. P. 191–195. DOI: 10.1090/S0002-9904-1971-12675-7
Bakhbukh M., Nikishin E. M. The convergence of the double Fourier series of continuous functions // Siberian Math. Journal, 1973. Vol. 14, no. 6. P. 832–839. DOI:10.1007/BF00975888
Bakhvalov A. N. Divergence everywhere of the Fourier series of continuous functions of several variables // Sb. Math., 1997. Vol. 188, no. 8. P. 1153–1170. DOI: 10.1070/SM1997v188n08ABEH000240
Bakhvalov A. N. \(\lambda \)-divergence of the Fourier series of continuous functions of several variables // Mathematical Notes, 2002. Vol. 72, no. 3–4. P. 454–465. DOI: 10.4213/mzm438
Antonov N.Yu. On divergence almost everywhere of Fourier series of continuous functions of two variables // Izvestiya of Saratov University. New ser. Ser. Math. Mech. Inform., 2014. Vol. 14, iss. 4, pt. 2. P. 495–502. [in Russian]
Stein E.M. On limits of sequences of operators // Ann. Math., 1961. Vol. 74, no. 1. P. 140–170.
Zygmund A. Trigonometric series, vol. 1. New York: Cambridge Univ. Press, 1959. 383 p
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