Celebratio Mathematica

I. C. Daubechies: Representation of quantum mechanical operators by kernels on Hilbert spaces of analytic functions. Ph.D. thesis, Vrije Universiteit Brussel, 1980. Advised by J. Reignier and A. Grossmann. phdthesis

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I. Daubechies and A. Grossmann: “An integral transform related to quantization,” J. Math. Phys. 21 : 8 (1980), pp. 2080–2090. MR 579204 article

I. Daubechies, A. Grossmann, and J. Reignier: “An integral transform related to quantization, II: Some mathematical properties,” J. Math. Phys. 24 : 2 (1983), pp. 239–254. MR 692298 article

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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.

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I. Daubechies and A. Grossmann: “Frames in the Bargmann space of entire functions,” Commun. Pure Appl. Math. 41 : 2 (1988), pp. 151–164. MR 924682 Zbl 0632.30049 article

We look at the decomposition of arbitrary \( f \) in \( L^2(R) \) in terms of the family of functions \[ \phi_{mn}(x) = \pi^{-1/4}\exp\bigl\{-\tfrac{1}{2}imnab + i\max - \tfrac{1}{2}(x-nb)^2\bigr\} ,\] with \( a,b > 0 \). We derive bounds and explicit formulas for the minimal expansion coefficients in the case where \( ab = 2\pi/N \), \( N \) an integer \( \geq 2 \). Transported to the Hilbert space \( F \) of entire functions introduced by V. Bargmann, these results are expressed as inequalities of the form \[ \mathbf{m}\|f\|^2 \leqq \sum_{m,n\in\mathbb{Z}} |f(z_{mn})|^2\exp\bigl\{-\tfrac{1}{2}|z_{mn}|^2\bigr\} \leqq \mathbf{M}\|f\|^2, \] where \( z_{mn} = ma + inb \), \( \mathbf{m} \), \( \textbf{M} > 0 \), and \( \|\cdot\| \) is the norm in \( F \), \[ \|f\|^2 = (2\pi)^{-1}\iint_{\mathbb{R}^2}dx\,dy\,|f(x+iy)|^2\exp\bigl\{-\tfrac{1}{2}(x^2 + y^2)\bigr\}. \] We conjecture that these inequalities remain true for all \( a,b \) such that \( ab < 2\pi \).

I. Daubechies: “Orthogonal bases of wavelets with finite support: Connection with discrete filters,” pp. 38–66 in Wavelets: Time-frequency methods and phase space (Marseille, France, 14–18 December 1987). Edited by J.-M. Combes, A. Grossmann, and P. Tchamitchian. Springer (Berlin), 1989. Zbl 0850.42013 incollection

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