Celebratio Mathematica — Daubechies

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Works connected to Augustus J. E. M. Janssen

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I. Daubechies and A. J. E. M. Janssen: “Two theorems on lattice expansions,” IEEE Trans. Inf. Theory 39 : 1 (1993), pp. 3–6. MR 1211486 Zbl 0764.42018 article

It is shown that there is a tradeoff between the smoothness and decay properties of the dual functions, occurring in the lattice expansion problem. More precisely, it is shown that if \( g \) and \( \tilde{g} \) are dual, then

at least one of \( H^{1/2}g \) and \( H^{1/2}\tilde{g} \) is not in \( L^2(\mathbb{R}) \), and
at least one of \( Hg \) and \( \tilde{g} \) is not in \( L^2(\mathbb{R}) \). Here, \( H \) is the operator \[ \frac{1}{4\pi^2} \frac{d^2}{dt^2} + t^2 .\]

The first result is a generalization of a theorem first stated by Balian and independently by Low, which was recently rigorously proved by Coifman and Semmes; a new, much shorter proof was very recently given by Battle. Battle suggests a theorem of type (1), but our result is stronger in the sense that certain implicit assumptions made by Battle are removed. Result (2) is new and relies heavily on the fact that, when \( G\in W^{2,2}(S) \) with \[ S = \bigl[-\tfrac12, \tfrac12\bigr]\times \bigl[-\tfrac12, \tfrac12\bigr] \] and \( G(0)=0 \), then \( 1/G\notin L^2(S) \). The latter result was not known to us and may be of independent interest.