  We study Toeplitz-type operators with respect to specific wavelets whose
Fourier transforms are related to Laguerre polynomials. On the one hand, this
choice of wavelets underlines the fact that these operators acting on wavelet
subspaces share many properties with the classical Toeplitz operators acting on
the Bergman spaces. On the other hand, it enables to study poly-Bergman spaces
and Toeplitz operators acting on them from a different perspective. Restricting
to symbols depending only on vertical variable in the upper half-plane of the
complex plane these operators are unitarily equivalent to a multiplication
operator with a certain function. Since this function is responsible for many
interesting features of these Toeplitz-type operators and their algebras, we
investigate its behavior in more detail. As a by-product we obtain an
interesting observation about the asymptotic behavior of true polyanalytic
Bergman spaces. Isomorphisms between the Calder\'on-Toeplitz operator algebras
and functional algebras are described and their consequences are discussed.
