×

zbMATH — the first resource for mathematics

On the uniqueness and integrability of multiple trigonometric series. (English. Russian original) Zbl 1192.42007
Math. Notes 86, No. 5, 716-728 (2009); translation from Mat. Zametki 86, No. 5, 761-775 (2009).
Summary: 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.

MSC:
42B05 Fourier series and coefficients in several variables
42B08 Summability in several variables
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. Bourgain, ”Spherical summation and uniqueness of multiple trigonometric series,” Internat. Math. Res. Notices, No. 3, 93–107 (1996). · Zbl 0852.42004
[2] J. M. Ash and G. Wang, ”Some spherical uniqueness theorems for multiple trigonometric series,” Ann. of Math. (2) 151(1), 1–33 (2000). · Zbl 0955.42010 · doi:10.2307/121110
[3] B. Connes, ”Sur les coefficients des séries trigonométriques convergentes sphériquement,” C. R. Acad. Sci. Paris Sér. A 283(4), 159–161 (1976).
[4] J. M. Ash, C. Freiling, and D. Rinne, ”Uniqueness of rectangularly convergent trigonometric series,” Ann. of Math. (2) 137(1), 145–166 (1993). · Zbl 0780.42015 · doi:10.2307/2946621
[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)]. · Zbl 0867.42007
[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)]. · Zbl 0772.42021 · doi:10.1070/SM1992v073n02ABEH002560
[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)]. · Zbl 0824.42006
[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)]. · Zbl 0639.42009
[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)]. · Zbl 0594.42008
[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)]. · Zbl 0696.42022
[11] S. Saks, Theory of the Integral (New York, 1939; Inostr. Lit., Moscow, 1949) [in Russian].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.