Celebratio Mathematica — Daubechies

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I. Daubechies, S. Jaffard, and J.-L. Journé: “A simple Wilson orthonormal basis with exponential decay,” SIAM J. Math. Anal. 22 : 2 (1991), pp. 554–572. An erratum to this article was published in SIAM J. Math. Anal. 22:3 (1991). MR 1084973 Zbl 0754.46016 article

Following a basic idea of Wilson [“Generalized Wannier functions”, preprint] orthonormal bases for \( L^2(\mathbb{R}) \) which are a variation on the Gabor scheme are constructed. More precisely, \( \phi \in L^2(\mathbb{R}) \) is constructed such that the \( \psi_{ln} \), \( l\in\mathbb{N} \), \( n\in\mathbb{Z} \), defined by \begin{align*} \psi_{0n} (x) &= \phi (x - n), \\ \psi_{ln} (x) &= \begin{cases} \sqrt 2\,\phi \bigl(x - n/2\bigr)\cos (2\pi lx) & l \ne 0,\ l+n \in 2\mathbb{Z}, \\ \sqrt 2\,\phi\bigl(x - n/2\bigr) \sin (2\pi lx) & l \ne 0,\ l+n \in 2\mathbb{Z}{+}1, \end{cases} \end{align*} constitute an orthonormal basis. Explicit examples are given in which both \( \phi \) and its Fourier transform \( \hat{\phi} \) have exponential decay. In the examples \( \phi \) is constructed as an infinite superposition of modulated Gaussians, with coefficients that decrease exponentially fast. It is believed that such orthonormal bases could be useful in many contexts where lattices of modulated Gaussian functions are now used.

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I. Daubechies, S. Jaffard, and J.-L. Journé: “Erratum: A simple Wilson orthonormal basis with exponential decay,” SIAM J. Math. Anal. 22 : 3 (1991), pp. 878. Erratum to article published in SIAM J. Math. Anal. 22:2 (1991). MR 1101314 Zbl 0794.46008 article

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