A Simple Wilson Orthonormal Basis with Exponential Decay

Source abstract

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 \[ \begin{gathered} \psi _{0n} (x) = \phi \left( {x - n} \right) \hfill \\ \psi _{in} (x) = \sqrt 2 \phi \left( {x - \frac{n} {2}} \right)\cos (2\pi lx)\,{\text{if}}\,l \ne 0,\,l + n \in 2\mathbb{Z} \hfill \\ = \sqrt 2 \phi \left( {x - \frac{n} {2}} \right)\sin (2\pi lx)\,{\text{if}}\,l \ne 0,\,l + n \in 2\mathbb{Z} + 1, \hfill \\ \end{gathered} \] 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.