Fractals and distributions on the 𝑁-torus

Shapiro, Victor

doi:10.1090/S0002-9939-03-06929-6

Fractals and distributions on the $N$-torus
HTML articles powered by AMS MathViewer

by Victor L. Shapiro PDF

Proc. Amer. Math. Soc. 131 (2003), 3431-3440 Request permission

Abstract:

This paper establishes non-Cartesian product sets, called fractal carpets and fractal foam, as sets of uniqueness for a class of trigonometric series.

References

J. M. Ash and G. Wang, Sets of uniqueness for spherically convergent multiple trigonometric series, preprint, 1999, 20 pages.
L. Bers, F. John, and M. Schechter, Partial Differential Equations, Interscience Publishers, New York, 1964.
Jean-Pierre Kahane and Raphaël Salem, Ensembles parfaits et séries trigonométriques, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1301, Hermann, Paris, 1963 (French). MR 0160065
Benoit B. Mandelbrot, The fractal geometry of nature, Schriftenreihe für den Referenten. [Series for the Referee], W. H. Freeman and Co., San Francisco, Calif., 1982. MR 665254
Raphaël Salem, Algebraic numbers and Fourier analysis, D. C. Heath and Company, Boston, Mass., 1963. MR 0157941
Victor L. Shapiro, Algebraic integers and distributions on the $N$-torus, J. Functional Analysis 13 (1973), 138–153. MR 0348392, DOI 10.1016/0022-1236(73)90041-4
Victor L. Shapiro, Sets of uniqueness on the $2$-torus, Trans. Amer. Math. Soc. 165 (1972), 127–147. MR 308684, DOI 10.1090/S0002-9947-1972-0308684-0
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776