Abstract
We obtain a sufficient condition for a set of measure zero in N-dimensional space to be a set of absolute convergence (A.C. set) for an N-tuple trigonometric series. We also show that, in a certain subclass of sets of measure zero (namely in the subclass of “ monotonic” curves), this condition cannot be sharpened.
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N. K. Bari, Trigonometric Series [in Russian], Fizmatgiz, Moscow (1961).
I. P. Natanson, The Theory of Functions of a Real Variable [in Russian], Moscow-Leningrad (1950).
R. A. Avetisyan, “On the sets of absolute convergence of double trigonometric series,” Matem. Zametki,11, No. 5, 473–480 (1972).
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Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 625–635, May, 1973.
The author wishes to thank E. M. Nikishin for some valuable remarks.
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Avetisyan, R.A. On sets of absolute convergence for multiple trigonometric series. Mathematical Notes of the Academy of Sciences of the USSR 13, 377–382 (1973). https://doi.org/10.1007/BF01147463
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DOI: https://doi.org/10.1007/BF01147463