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Victor Shapiro and the Theory of Uniqueness for Multiple Trigonometric Series

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Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1)

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 4))

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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.

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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.

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Authors and Affiliations

  1. Mathematics Department, DePaul University Chicago, Chicago, IL, 60614, USA

    J. Marshall Ash

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  1. J. Marshall Ash
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Correspondence to J. Marshall Ash .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico, USA

    María Cristina Pereyra

  2. Department of Mathematics, Venezuelan Institute for Scientific Research, Caracas, Venezuela

    Stefania Marcantognini

  3. Department of Mathematical Sciences, Georgia Southern University, Statesboro, Georgia, USA

    Alexander M. Stokolos

  4. Department of Mathematics and Actu rial Science, Roosevelt University, Chicago, Illinois, USA

    Wilfredo Urbina

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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

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