A Study of Offset Linear Canonical Wavelet Transforms on Certain Function Spaces

This article introduces a function space \(\mathcal {S}_{\textbf{A}}(\mathbb {R})\) ${\mathcal{S}}_{\mathbf{A}}(\mathbb{R})$ in the offset linear canonical domain, which is a counterpart of the Schwartz space \(\mathcal {S}(\mathbb {R})\) $\mathcal{S}(\mathbb{R})$ in the Fourier domain, and studies the properties of offset linear canonical transform (OLCT) on this space. Additionally, the paper introduces a new definition of offset linear canonical wavelet transform (OLCWT) which is based on the convolution associated with the OLCT. Several properties of OLCWT on \(\mathcal {S}_{\textbf{A}}(\mathbb {R})\) ${\mathcal{S}}_{\mathbf{A}}(\mathbb{R})$ are investigated. Using these properties, we eventually establish that the OLCWT is a continuous linear operator on some newly defined test function spaces \(\mathfrak {A}_P^\textbf{A}\) ${\mathfrak{A}}_P^{\mathbf{A}}$ and \(\textrm{B}_\textbf{A}(\mathbb {R}^2)\) ${\text{B}}_{\mathbf{A}}({\mathbb{R}}^2)$ .