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The linear canonical wavelet transform on some function spaces

Yong Guo

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China

Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, P. R. China

E-mail Address: [email protected]

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and

Bing-Zhao Li

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P. R. China

Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, P. R. China

E-mail Address: [email protected]

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https://doi.org/10.1142/S0219691318500108Cited by:46 (Source: Crossref)

Abstract

It is well known that the domain of Fourier transform (FT) can be extended to the Schwartz space $𝒮 (R)$ 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 $𝒮 (R)$ . Therefore, a space $𝒮_{A_{1}} (R)$ generalized from $𝒮 (R)$ is introduced firstly, and further we prove that LCT is a homeomorphism from $𝒮_{A_{1}} (R)$ 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 $𝒮_{A_{1}} (R)$ . Based on these properties, we finally prove that LCWT is a linear continuous operator on the spaces of $L^{p, A_{1}}$ and $H_{A_{1}}^{s, p}$ .

Keywords:

AMSC: 46F12, 46E35, 46A11

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