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Abstract: Let $G$ be the ``$ax+b$''-group with the left invariant Haar measure $d\nu$ and $\psi$ be a fixed real-valued admissible wavelet on $L_2(\mathbb{R})$. The structure of the space of Calder\'on (wavelet) transforms $W_{\psi}(L_{2}(\mathbb{R}))$ inside $L_{2}(G,d\nu)$ is described after identifying the group $G$ with the upper half-plane $\Pi$ in $\mathbb{C}$. Using this result some properties and the Wick calculus of the Calder\'on-Toeplitz operators $T_{a}$ acting on $W_{\psi}(L_{2}(\mathbb{R}))$ whose symbols $a=a(\zeta)$ depend on $v=\Im \zeta$ for $\zeta \in G$ are investigated. Contact the authors: ondrej.hutnik@upjs.sk Download PDF version of the preprint. |