A Cantor-Lebesgue theorem in two dimensions
HTML articles powered by AMS MathViewer
- by Roger Cooke PDF
- Proc. Amer. Math. Soc. 30 (1971), 547-550 Request permission
Abstract:
We prove a partial extension of the classical result of Cantor and Lebesgue to two dimensions, for circular summation of trigonometric series. We assume convergence almost everywhere rather than merely on a set of positive measure. Although the theorem we prove has applications in uniqueness theorems, its purpose is to suggest what may be true in general, since the proof does not generalize to higher dimensions.References
- Min-Teh Cheng, Uniqueness of multiple trigonometric series, Ann. of Math. (2) 52 (1950), 403–416. MR 36859, DOI 10.2307/1969476
- Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869
- 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
- A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 30 (1971), 547-550
- MSC: Primary 42.10
- DOI: https://doi.org/10.1090/S0002-9939-1971-0282134-X
- MathSciNet review: 0282134