Two-dimensional Shannon type expansions via one-dimensional affine and wavelet lattice actions

Abstract: It is rather unexpected, but true, that it is possible to construct reproducing formulae and orthonormal bases of $L^2 (\mathbb{R}^2)$ just by applying the standard one dimensional wavelet action of translations and dilations to the first variable $x_1$ of the generating function $\psi(x_1,x_2)$, $\psi \in L^2 (\mathbb{R}^2)$, i.e., by making use of building blocks $$\psi_{u,s}(x_1,x_2)=s^{-1/2}\psi\left(\frac{x_1-u}{s},x_2\right), \text{where } u\in \mathbb{R}, s>0,$$ in the case of reproducing formulae, and $$\psi_{k,m}(x_1,x_2)=2^{-k/2} \psi\left(\frac{x_1-2^k m}{2^k},x_2 \right), \text{where } k,m\in \mathbb{Z},$$ in the case of orthonormal bases. It is possible to compensate the fact, that the second variable $x_2$ is not acted upon, by a careful selection of the generating function $\psi$. Shannon wavelet tiling of the time-frequency plane $\mathbb{R}^2$, a standard illustration of orthogonality and completeness phenomena corresponding to the Shannon wavelet, $$ \chi_{(2^km,2^k(m+1)]}(x) \chi_{2^{-k}I}(\xi),\, k,m\in \mathbb{Z}, \,I=- (1/2,1]\cup (1/2,1], $$ with $x$ representing time and $\xi$ frequency, is substituted by a phase space tiling of $\mathbb{R}^4$ with unbounded, hyperboloid type blocks of the form $$ \chi_{(2^km,2^k(m+1)]}(x_1)\sum_{n,l}\chi_{2^{-k}I_{D(n,l)}}(\xi_1) \chi_{(n,n+1]}(x_2)\chi_{(l,l+1]}(\xi_2),\, k,m\in \mathbb{Z} $$ where $I_r=2^{-r}I$, $ r\ge 1$, and $D:\mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{N}$ is a bijection, an additional parameter of the generating function, needed for the lift from $L^2(\mathbb{R})$ to $L^2(\mathbb{R}^2)$. Variables $x_1,x_2$ are coordinates of position and variables $\xi_1,\xi_2$ of momentum.