On the divergence of triangular and eccentric spherical sums of double Fourier series
Sbornik. Mathematics, Tome 207 (2016) no. 1, pp. 65-84

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We construct a continuous function on the torus with almost everywhere divergent triangular sums of double Fourier series. We also prove an analogous theorem for eccentric spherical sums. Bibliography: 14 titles.
Keywords: divergent triangular sums, double Fourier series.
G. A. Karagulyan. On the divergence of triangular and eccentric spherical sums of double Fourier series. Sbornik. Mathematics, Tome 207 (2016) no. 1, pp. 65-84. http://geodesic.mathdoc.fr/item/SM_2016_207_1_a2/
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[1] L. Carleson, “On convergence and growth of partial sums of Fourier series”, Acta Math., 116 (1966), 135–157 | DOI | MR | Zbl

[2] R. A. Hunt, “On the convergence of Fourier series”, Orthogonal expansions and their continuous analogues (Edwardsville, Ill., 1967), Southern Illinois Univ. Press, Carbondale, Ill., 1968, 235–255 | MR | Zbl

[3] P. Sjölin, “Convergence almost everywhere of certain singular integrals and multiple Fourier series”, Ark. Mat., 9 (1971), 65–90 | DOI | MR | Zbl

[4] N. Yu. Antonov, “Convergence of Fourier series”, East J. Approx., 2:2 (1996), 187–196 | MR | Zbl

[5] C. Fefferman, “On the convergence of multiple Fourier series”, Bull. Amer. Math. Soc., 77:5 (1971), 744–745 | DOI | MR | Zbl

[6] N. R. Tevzadze, “O skhodimosti dvoinogo ryada Fure funktsii, summiruemoi s kvadratom”, Soobsch. AN Gruz. SSR, 58:2 (1970), 277–279 | MR | Zbl

[7] C. Fefferman, “On the divergence of multiple Fourier series”, Bull. Amer. Math. Soc., 77:2 (1971), 191–195 | DOI | MR | Zbl

[8] G. A. Karagulyan, K. R. Muradyan, “Divergent triangular sums of double Fourier series”, J. Contemp. Math. Anal., 50:4 (2015), 196–207 | DOI

[9] G. A. Karagulyan, K. R. Muradyan, “On the divergence of Walsh and Haar series by sectorial and triangular regions”, Proc. Yerevan State Univ. Phys. Math. Sci., 234:2 (2014), 3–12 | Zbl

[10] G. A. Karagulyan, “On unboundedness of maximal operators for directional Hilbert transforms”, Proc. Amer. Math. Soc., 135:10 (2007), 3133–3141 | DOI | MR | Zbl

[11] A. Zygmund, Trigonometric series, v. I, 2nd ed., Cambridge Univ. Press, New York, 1959, xii+383 pp. | MR | MR | Zbl | Zbl

[12] E. M. Nikishin, P. L. Ulyanov, “Ob absolyutnoi i bezuslovnoi skhodimosti”, UMN, 22:3(135) (1967), 240–242 | MR

[13] B. S. Kashin, A. A. Saakyan, Orthogonal series, Transl. Math. Monogr., 75, Amer. Math. Soc., Providence, RI, 1989, xii+451 pp. | MR | MR | Zbl | Zbl

[14] J. Neveu, Bases mathématiques du calcul des probabilités, Masson et Cie, Éditeurs, Paris, 1964, xiii+203 pp. | MR | MR | Zbl | Zbl