Sets of uniqueness on the 2-torus

Shapiro, Victor L.

doi:10.1090/S0002-9947-1972-0308684-0

Sets of uniqueness on the $2$-torus
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Trans. Amer. Math. Soc. 165 (1972), 127-147 Request permission

Abstract:

${H^{(J)}}$-sets are defined on the 2-torus and the following results are established: (1) ${H^{(J)}}$-sets are sets of uniqueness both for Abel summability and circular convergence of double trigonometric series; (2) a countable union of closed sets of uniqueness of type (A) (i.e., Abel summability) is also a set of uniqueness of type (A).

References

Leonard D. Berkovitz, Circular summation and localization of double trigonometric series, Trans. Amer. Math. Soc. 70 (1951), 323–344. MR 40464, DOI 10.1090/S0002-9947-1951-0040464-2

A Cantor-Lebesgue theorem for two dimensions

17

Roger Cooke, A Cantor-Lebesgue theorem in two dimensions, Proc. Amer. Math. Soc. 30 (1971), 547–550. MR 282134, DOI 10.1090/S0002-9939-1971-0282134-X

Subharmonic functions

Theorie de l’integrale

29

Victor L. Shapiro, Fourier series in several variables, Bull. Amer. Math. Soc. 70 (1964), 48–93. MR 158222, DOI 10.1090/S0002-9904-1964-11026-0
Victor L. Shapiro, Uniqueness of multiple trigonometric series, Ann. of Math. (2) 66 (1957), 467–480. MR 90700, DOI 10.2307/1969904
Victor L. Shapiro, Localization of conjugate multiple trigonometric series, Ann. of Math. (2) 61 (1955), 368–380. MR 68027, DOI 10.2307/1969919
Victor L. Shapiro, The divergence theorem for discontinuous vector fields, Ann. of Math. (2) 68 (1958), 604–624. MR 100725, DOI 10.2307/1970158
Victor L. Shapiro, Removable sets for pointwise solutions of the generalized Cauchy-Riemann equations, Ann. of Math. (2) 92 (1970), 82–101. MR 437898, DOI 10.2307/1970698
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776