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On the uniqueness and integrability of multiple trigonometric series

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Abstract

Under minimal constraints on the coefficients, we prove uniqueness theorems for multiple trigonometric series in which, instead of pointwise convergence, we consider the convergence of integral means of spherical, cubic, and other partial sums. We also obtain sufficient conditions for the integrability of multiple trigonometric series, i.e., conditions under which these series are Fourier series.

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References

  1. J. Bourgain, “Spherical summation and uniqueness of multiple trigonometric series,” Internat. Math. Res. Notices, No. 3, 93–107 (1996).

  2. J. M. Ash and G. Wang, “Some spherical uniqueness theorems for multiple trigonometric series,” Ann. of Math. (2) 151(1), 1–33 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  3. B. Connes, “Sur les coefficients des séries trigonométriques convergentes sphériquement,” C. R. Acad. Sci. Paris Sér. A 283(4), 159–161 (1976).

    MATH  MathSciNet  Google Scholar 

  4. J. M. Ash, C. Freiling, and D. Rinne, “Uniqueness of rectangularly convergent trigonometric series,” Ann. of Math. (2) 137(1), 145–166 (1993).

    Article  MathSciNet  Google Scholar 

  5. Sh. T. Tetunashvili, “On the uniqueness of multiple trigonometric series,” Mat. Zametki 58(4), 596–603 (1995) [Math. Notes 58 (3–4), 1094–1099 (1995)].

    MathSciNet  Google Scholar 

  6. Sh. T. Tetunashvili, “On some multiple function series and the solution of the uniqueness problem for Pringsheim convergence of multiple trigonometric series,” Mat. Sb. 73(2), 517–534 (1991) [Math. USSRSb. 182 (8), 1158–1176 (1991)].

    Article  MATH  MathSciNet  Google Scholar 

  7. A. A. Talalyan, “On some uniqueness properties of multiple trigonometric series and harmonic functions,” Izv. Akad. Nauk SSSR Ser. Mat. 52(3), 621–650 (1988) [Math. USSR-Izv. 32 (3), 627–654 (1988)].

    MathSciNet  Google Scholar 

  8. A. A. Talalyan, “On the uniqueness of multiple trigonometric series,” Mat. Sb. 132(1), 104–130 (1987) [Math. USSR-Sb. 60 (1), 107–131 (1988)].

    Google Scholar 

  9. A. A. Talalyan, “On the uniqueness of double trigonometric series,” Izv. Akad. Nauk Armyan. SSR Ser. Mat. 20(6), 426–462 (1985) [Sov. J. Contemp.Math. Anal., Arm. Acad. Sci. 20 (6), 20–58 (1985)].

    MathSciNet  Google Scholar 

  10. V. A. Skvortsov and A. A. Talalyan, “Some problems of the uniqueness of multiple series in the Haar system and a trigonometric system,” Mat. Zametki 46(2), 104–113 (1989) [Math. Notes 46 (1–2), 646–653 (1990)].

    MATH  MathSciNet  Google Scholar 

  11. S. Saks, Theory of the Integral (New York, 1939; Inostr. Lit., Moscow, 1949) [in Russian].

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Armenia, Yerevan, Armenia

    A. A. Talalyan

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  1. A. A. Talalyan
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Correspondence to A. A. Talalyan.

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Original Russian Text © A. A. Talalyan, 2009, published in Matematicheskie Zametki, 2009, Vol. 86, No. 5, pp. 761–775.

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Talalyan, A.A. On the uniqueness and integrability of multiple trigonometric series. Math Notes 86, 716–728 (2009). https://doi.org/10.1134/S0001434609110145

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