The continuous quaternion wavelet transform on distribution spaces

This article provides a revised version of some existing results in the literature for the quaternion Fourier transform (QFT) and quaternion wavelet transforms. The inner-product relation and its consequent formula for the continuous quaternion wavelet transform (CQWT) are derived in Lp(R2;H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>p ({\mathbb {R}}<^>{2}; {\mathbb {H}})$$\end{document} space under the assumption that the admissible wavelet is complex-valued and has a real QFT. Furthermore, the characterization of quaternion Sobolev spaces Hs(R2;H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>{s}({\mathbb {R}}<^>{2}; {\mathbb {H}})$$\end{document} and Wm,p(Omega;H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W<^>{m,p} (\Omega ; {\mathbb {H}})$$\end{document}, weighted quaternion Sobolev space Wkm,p(Omega;H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{k}<^>{m,p} (\varvec{\Omega }; {\mathbb {H}} )$$\end{document} and generalized quaternion Sobolev space Hw omega(R2;H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{w}<^>{\omega } ({\mathbb {R}}<^>{2}; {\mathbb {H}})$$\end{document}, quaternion Besov space by means of the CQWT is presented. The CQWT is analysed within these function and distribution spaces, yielding novel findings regarding continuity and boundedness.