The linear canonical wavelet transform on some function spaces

It is well known that the domain of Fourier transform (FT) can be extended to the Schwartz space (R) for convenience. As a generation of FT, it is necessary to detect the linear canonical transform (LCT) on a new space for obtaining the similar properties like FT on (R). Therefore, a space A1(R) generalized from (R) is introduced firstly, and further we prove that LCT is a homeomorphism from A1(R) onto itself. The linear canonical wavelet transform (LCWT) is a newly proposed transform based on the convolution theorem in LCT domain. Moreover, we propose an equivalent definition of LCWT associated with LCT and further study some properties of LCWT on A1(R). Based on these properties, we finally prove that LCWT is a linear continuous operator on the spaces of Lp,A1 and HA1s,p.