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).
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
- Measure Zero
- Trigonometric Series
- Absolute Convergence
- Multiple Trigonometric Series