Properties of the affine Poincaré wavelet transform

The affine Poincaré wavelet transform is the convolution of the analyzed function and the parameter-dependent function called a wavelet. The wavelet is constructed from a function called the mother wavelet by using Lorentz transformations, shift and scaling and depending on the parameters, characterizing these transformations. We provide uniform by parameters estimates for the affine Poincaré wavelet transforms in some classes of analyzed functions and mother wavelets. Among other things, an estimate of the transform for large shifts and an estimate for large rapidities is proved. Both estimates allow one to check vanishment of the transform at small scales.
We provide an asymptotic calculation of the Poincare affine wavelet transform of the model functions.