Approximation of linear canonical wavelet transform on the generalized Sobolev spaces (Q2280114)

The introduction is well written and clearly explains the motivation of the authors and the environment in which the work takes place; one can read: Many fundamental results of the LCWT are already known, but the approximation properties of the LCWT on the generalized Sobolev spaces $B^{\xi, A}_{p,k}(\mathbb{R})$ are still missing. (The definition of these generalized Sobolev spaces is given page 860 (Definition 3.3).) So the aim of the authors is clear. A part of the abstract is the following: The main objective of this paper is to study the linear canonical wavelet transform (LCWT) on the generalized Sobolev space $B^{\xi, A}_{p,k}(\mathbb{R})$ and generalized weighted space $L^{s,p}_{\varepsilon, A}(\mathbb{R})$. Its approximation properties and convergence of convolution for $F^{A}_{\psi}$ in the space $B^{\xi, A}_{p,k}(\mathbb{R})$ are also discussed. [\dots] Continuity and boundedness of the LCWT on several spaces and composition of LCWTs are also studied. The theorems are quite technical but are well presented throughout the paper. The results, using a generalisation of the classical wavelet transform, are not only interesting for specialists of the specific aspects treated (see the introduction) but also for people involved in wavelet theory in general.