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A simple Wilson orthonormal basis with exponential decay. (English) Zbl 0754.46016

Summary: 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 \(\psi_{\ell n}\), \(\ell\in\mathbb{N}\), \(n\in\mathbb{Z}\), defined by \[ \begin{aligned} \psi_{0n}(x) &=\phi(x- n)\\ \psi_{\ell n}(x) &= \sqrt{2}\phi\left( x-{n\over 2}\right)\cos(2\pi\ell x) \qquad\text{if } \ell\neq 0,\;\ell+n\in 2\mathbb{Z} \qquad \text{ and }\\ &= \sqrt{2}\phi\left( x-{n\over 2}\right)\sin(2\pi\ell x) \qquad\text{if } \ell\neq 0,\;\ell+n\in2\mathbb{Z}+1,\end{aligned} \] consitute 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.

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
42C20 Other transformations of harmonic type
94A11 Application of orthogonal and other special functions
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