Abstract
In 1870, Georg Cantor proved that if a trigonometric series converges to 0 everywhere, then all its coefficients must be 0. In the twentieth century this result was extended to higher dimensional trigonometric series when the mode of convergence is taken to be spherical convergence and also when it is taken to be unrestricted rectangular convergence. We will describe the path to each result. An important part of the first path was Victor Shapiro’s seminal 1957 paper, Uniqueness of multiple trigonometric series. This paper also was an unexpected part of the second path.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
J.M. Ash, A new, harder proof that continuous functions with Schwarz derivative 0 are lines, in Contemporary Aspects of Fourier Analysis, ed. by W.O. Bray, C.V. Stanojevic, Pure and Appl. Math. vol. 157 (1994). Marcel Dekker, New York, pp. 35–46
J.M. Ash, Uniqueness for Spherically Convergent Multiple Trigonometric Series, ed. by G. Anastassiou. Handbook of Analytic-Computational Methods in Applied Mathematics (Chapman & Hall/CRC, Boca Raton, 2000), pp. 309–355
J.M. Ash, Uniqueness for higher dimensional trigonometric series. Cubo 4, 97–125 (2002)
J.M. Ash, C. Freiling, D. Rinne, Uniqueness of rectangularly convergent trigonometric series. Ann. Math. 137, 145–166 (1993)
J.M. Ash, G. Wang, A survey of uniqueness questions in multiple trigonometric series, in A Conference in Harmonic Analysis and Nonlinear Differential Equations in Honor of Victor L. Shapiro, vol. 208. Contemporary Mathematics (1997), pp. 35–71
J.M. Ash, G. Wang, One and two dimensional Cantor-Lebesgue type theorems. Trans. Am. Math. Soc. 349, 1663–1674 (1997)
J.M. Ash, G.V. Welland, Convergence, uniqueness, and summability of multiple trigonometric series. Trans. Am. Math. Soc. 163, 401–436 (1972). MR 45 # 9057
J. Bourgain, Spherical summation and uniqueness of multiple trigonometric series. Int. Math. Res. Not. 1996, 93–107 (1996)
G. Cantor, Beweis, das eine fur jeden reellen Wert von x durch eine trigonometrische Reihe gegebene Funktion f(x) sich nur auf eine einzige Weise in dieser Form darstellen lasst, Crelles J. fuir Math. 72, 139–142 (1870); also in Georg Olms, Gesammelte Abhandlungen, Hildesheim, 1962, pp. 80–83
P.J. Cohen, Topics in the theory of uniqueness of trigonometrical series, doctoral thesis, University of Chicago, Chicago, IL (1958)
B. Connes, Sur les coefficients des séries trigonometriques convergents sphériquement. CRA Sc Paris t. Ser. A 283, 159–161 (1976)
R.L. Cooke, A Cantor-Lebesgue theorem in two dimensions. Proc. Am. Math. Soc. 30, 547–550 (1971). MR 43 #7847
Mathematics Genealogy Project. http://genealogy.math.ndsu.nodak.edu/
V.L. Shapiro, Uniqueness of multiple trigonometric series. Ann. Math. 66, 467–480 (1957). MR 19, 854; 1432
S. Tetunashvili, On some multiple function series and the solution of the uniqueness problem for Pringsheim convergence of multiple trigonometric series. Mat. Sb. 182, 1158–1176 (1991) (Russian), Math. USSR Sbornik 73, 517–534 (1992) (English). MR 93b:42022
A. Zygmund, Trigonometric Series, 2nd edn. vol. 1 (Cambridge University Press, Cambridge, 1958)
A. Zygmund, A Cantor-Lebesgue theorem for double trigonometric series. Stud. Math. 43, 173–178 (1972). MR 47 #711
Acknowledgements
This research was partially supported by a grant from the Faculty and Development Program of the College of Liberal Arts and Sciences, DePaul University.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Ash, J.M. (2016). Victor Shapiro and the Theory of Uniqueness for Multiple Trigonometric Series. In: Pereyra, M., Marcantognini, S., Stokolos, A., Urbina, W. (eds) Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1). Association for Women in Mathematics Series, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-30961-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-30961-3_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-30959-0
Online ISBN: 978-3-319-30961-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)