Celebratio Mathematica

I. Daubechies, A. Grossmann, and Y. Meyer: “Painless nonorthogonal expansions,” J. Math. Phys. 27 (1986), pp. 1271–1283. MR 836025 Zbl 0608.46014 article

In a Hilbert space \( \mathscr{H} \), discrete families of vectors \( \{h_j\} \) with the property that \[ f = \sum_j\langle h_j | f\rangle\,h_j \] for every \( f \) in \( \mathscr{H} \) are considered. This expansion formula is obviously true if the family is an orthonormal basis of \( \mathscr{H} \), but also can hold in situations where the \( h_j \) are not mutually orthogonal and are “overcomplete.” The two classes of examples studied here are

appropriate sets of Weyl–Heisenberg coherent states, based on certain (non-Gaussian) fiducial vectors, and
analogous families of affine coherent states.

It is believed, that such “quasiorthogonal expansions” will be a useful tool in many areas of theoretical physics and applied mathematics.

Yves F. Meyer	Related
Alexander Grossmann	Related

Wavelets and their applications. Edited by M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, and L. Raphael. Jones and Bartlett (Boston, MA), 1992. Zbl 0782.00087 book

Mary Beth Ruskai	Related
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Ronald Raphael Coifman	Related
Gregory Beylkin	Related
Louise Arakelian Raphael	Related
Stéphane Georges Mallat	Related

I. Daubechies: “Wavelets on the interval,” pp. 95–107 in Progress in wavelet analysis and applications: Proceedings of the 3rd international conference on wavelets and applications (Toulouse, France, 8–13 June 1992). Edited by Y. Meyer and S. Roques. Editions Frontières (Gif-sur-Yvette, France), 1993. MR 1282934 Zbl 0925.42017 incollection

Sylvie Roques	Related
Yves F. Meyer	Related

I. Daubechies: “The work of Yves Meyer,” pp. 115–124 in Proceedings of the International Congress of Mathematicians (Hyderabad, India, 19–27 August 2010), vol. 1. Edited by R. Bhatia, P. A. Gastesi, A. Pal, G. Rangarajan, V. Srinivas, and M. Vanninathan. Hindustan Book Agency (New Delhi), 2010. MR 2827886 Zbl 1225.01059 incollection

Yves Meyer has made numerous contributions to mathematics, several of which will be reviewed here, in particular in number theory, harmonic analysis and partial differential equations.

His work in harmonic analysis led him naturally to take an interest in wavelets, when they emerged in the early 1980s; his synthesis of the advanced theoretical results in singular integral operator theory, established by himself and others, and of the requirements imposed by practical applications, led to enormous progress for wavelet theory and its applications. Wavelets and wavelet packets are now standard, extremely useful tools in many disciplines; their success is due in large measure to the vision, the insight and the enthusiasm of Yves Meyer.

Yves F. Meyer	Related
Vasudevan Srinivas	Related
Rajendra Bhatia	Related
Muthusamy Vanninathan	Related
Pablo Arés Gastesi	Related
Govindan Rangarajan	Related
Arup Kumar Pal	Related

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