Yves Meyer has been selected to win the 2017 Abel Prize. The citation is “for his pivotal role in the development of the mathematical theory of wavelets”. His work is certainly well known within mathematics, especially within harmonic analysis and in its important applications in image processing, data compression, signal analysis, and many other modern settings.
There are announcements of the prize in various places:
I will defer to these other sources for general information about the prize and about Meyer’s work. Here I would like to bring out a few aspects of his work with the help of Mathematical Reviews. First of all, at the time of this writing, there are 8448 items in MathSciNet with the word “wavelet” or “wavelets” in the title. Looking for either “wavelet” or “wavelets” anywhere in our records for items produces 13705 matches. The earliest is
MR0001894 Baker, Bevan B.; Copson, E. T. The Mathematical Theory of Huygens’ Principle. Oxford University Press, New York, (1939). vii+155 pp.
where “spherical wavelets” are mentioned in the review. (I suspect that these are not the same thing we normally think of as wavelets.)
Secondly, since citations are a big thing these days, let me point out that in MathSciNet, Meyer is cited 4834 times by 3262 authors. His most highly cited work is, not surprisingly, about wavelets:
MR1228209 Meyer, Yves. Wavelets and operators. Translated from the 1990 French original by D. H. Salinger. Cambridge Studies in Advanced Mathematics, 37. Cambridge University Press, Cambridge, 1992. xvi+224 pp. ISBN: 0-521-42000-8; 0-521-45869-2
This is the first part of a multi-part book. The second part was published as
MR1456993 Meyer, Yves; Coifman, Ronald. Wavelets. Calderón-Zygmund and multilinear operators. Translated from the 1990 and 1991 French originals by David Salinger. Cambridge Studies in Advanced Mathematics, 48. Cambridge University Press, Cambridge, 1997. xx+315 pp. ISBN: 0-521-42001-6; 0-521-79473-0
The book was published in two parts in English, but in three parts in French:
MR1085487 Meyer, Yves. Ondelettes et opérateurs. I. Ondelettes. Actualités Mathématiques. Hermann, Paris, 1990.
MR1085488 Meyer, Yves. Ondelettes et opérateurs. II. Opérateurs de Calderón-Zygmund. Actualités Mathématiques. Hermann, Paris, 1990.
MR1160989 Meyer, Yves; Coifman, R. R. Ondelettes et opérateurs. III. Opérateurs multilinéaires. Actualités Mathématiques. Hermann, Paris, 1991
Meyer and Coifman have 37 joint publications listed in MathSciNet. The earliest is a paper on singular integrals published in the Transactions of the AMS. Their most frequently cited joint work is on pseudodifferential operators, and was published as a volume in the esteemed series Astérisque from the Société Mathématique de France.
Meyer has published three books with the American Mathematical Society:
MR1342019 Jaffard, Stéphane; Meyer, Yves. Wavelet methods for pointwise regularity and local oscillations of functions. Mem. Amer. Math. Soc. 123 (1996), no. 587, x+110 pp.
MR1483896 Meyer, Yves. Wavelets, vibrations and scalings. With a preface in French by the author. CRM Monograph Series, 9. American Mathematical Society, Providence, RI, 1998. x+133 pp. ISBN: 0-8218-0685-8
MR1852741 Meyer, Yves. Oscillating patterns in image processing and nonlinear evolution equations. The fifteenth Dean Jacqueline B. Lewis memorial lectures. University Lecture Series, 22. American Mathematical Society, Providence, RI, 2001. x+122 pp. ISBN: 0-8218-2920-3
There was a story last year about a recent result of Meyer’s. Meyer was looking at variations on the Poisson formula as given in the work of Nir Lev and Alexander Olevskii, Quasicrystals with discrete support and spectrum. Rev. Mat. Iberoam. 32 (2016), no. 4, 1341–1352 [MR3593527]. After lecturing several times on the result, Meyer came up with a simpler proof of the result. Like any good researcher, before sending off the paper to a journal, he checked the existing literature. In the references to the Lev and Olevskii paper, he found a paper by Guinand:
MR0107784 Guinand, A. P. Concordance and the harmonic analysis of sequences. Acta Math. 101 1959 235–271.
Meyer dug up a copy of the paper and was surprised to find that Guinand had the same solution as his own. But no one had noticed this. Lev and Olevskii had not. Nor, apparently, had Salomon Bochner, who made no mention of it in his review of the paper in Mathematical Reviews. Meyer adjusted his paper accordingly, giving priority to Guinand, and it was published in the Proceedings of the National Academy of Science USA:
MR3482845 Meyer, Yves F. Measures with locally finite support and spectrum. Proc. Natl. Acad. Sci. USA 113 (2016), no. 12, 3152–3158.
In the section labeled Significance, Meyer wrote, “Our new Poisson’s formulas were hidden inside an old and almost forgotten paper published in 1959 by A. P. Guinand.”
Wavelet image by JonMcLoone – Own work, CC BY-SA 3.0, Link
