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

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 $ψ(x_1,x_2)$, $ψ\in L^2 (\mathbb{R}^2)$, i.e., by making use of building blocks $ψ_{u,s}(x_1,x_2)=s^{-1/2}ψ\left(\frac{x_1-u}{s},x_2\right), \text{where } u\in \mathbb{R}, s>0,$ in the case of reproducing formulae, and $ψ_{k,m}(x_1,x_2)=2^{-k/2} ψ\left(\frac{x_1-2^k m}{2^k},x_2 \right), \text{where } k,m\in \mathbb{Z},$ 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 $ψ$. Shannon wavelet tiling of the time-frequency plane $\mathbb{R}^2$, a standard illustration of orthogonality and completeness phenomena corresponding to the Shannon wavelet, $χ_{(2^km,2^k(m+1)]}(x) χ_{2^{-k}I}(ξ),\, k,m\in \mathbb{Z}, \,I=- (1/2,1]\cup (1/2,1],$ with $x$ representing time and $ξ$ frequency, is substituted by a phase space tiling of $\mathbb{R}^4$ with unbounded, hyperboloid type blocks of the form $χ_{(2^km,2^k(m+1)]}(x_1)\sum_{n,l}χ_{2^{-k}I_{D(n,l)}}(ξ_1) χ_{(n,n+1]}(x_2)χ_{(l,l+1]}(ξ_2),\, k,m\in \mathbb{Z}$ 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 $ξ_1,ξ_2$ of momentum.

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