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Streszczenie

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 $$ \psi_{u,s}(x_1,x_2)=s^{-1/2}\psi\bigg(\frac{x_1-u}{s},x_2\bigg), \quad\ \text{where } u\in \mathbb R,\, s \gt 0, $$ in the case of reproducing formulae, and $$ \psi_{k,m}(x_1,x_2)=2^{-k/2} \psi\bigg(\frac{x_1-2^k m}{2^k},x_2 \bigg), \quad\ \text{where } k,m\in \mathbb Z, $$ 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 $\mathbb R^2$ is a standard illustration of orthogonality and completeness phenomena corresponding to the Shannon wavelet, $$ \chi_{(2^km,2^k(m+1)]}(x) \chi_{2^{-k}I}(\xi),\quad\ k,m\in \mathbb Z, \,I=- (1/2,1]\cup (1/2,1], $$ with $x$ representing time and $\xi$ frequency. In our current context, of the wavelet action restricted to the first coordinate of $\mathbb R^2$, it is substituted by a phase space tiling of $\mathbb R^4$ with unbounded, hyperboloid type blocks of the form $$ \chi_{(2^km,2^k(m+1)]}(x_1)\sum_{n,l}\chi_{2^{-k}I_{D(n,l)}}(\xi_1) \chi_{(n,n+1]}(x_2)\chi_{(l,l+1]}(\xi_2),\quad k,m\in \mathbb Z, $$ where $I_r=2^{-r}I$, $ r\ge 1$, and $D:\mathbb Z \times \mathbb Z \rightarrow \N$ is a bijection, an additional parameter of the generating function, needed for the lift from $L^2(\mathbb R)$ to $L^2(\mathbb R^2)$. The variables $x_1,x_2$ are the coordinates of position, and the variables $\xi_1,\xi_2$ of momentum.