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It is rather unexpected, but true, that it is possible to constructreproducing formulae and orthonormal bases of $L^2 (\mathbb{R}^2)$ just byapplying the standard one dimensional wavelet action of translations anddilations 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 buildingblocks $$\psi_{u,s}(x_1,x_2)=s^{-1/2}\psi\left(\frac{x_1-u}{s},x_2\right),\text{where } u\in \mathbb{R}, s>0,$$ 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 ispossible to compensate the fact, that the second variable $x_2$ is not actedupon, by a careful selection of the generating function $\psi$. Shannon wavelettiling of the time-frequency plane $\mathbb{R}^2$, a standard illustration oforthogonality 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, issubstituted 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.

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