Dilation-and-modulation systems on the half real line

Translation, dilation, and modulation are fundamental operations in wavelet analysis. Affine frames based on translation-and-dilation operation and Gabor frames based on translation-and-modulation operation have been extensively studied and seen great achievements. But dilation-and-modulation frames have not. This paper addresses a class of dilation-and-modulation systems in L2(R+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}(\mathbb {R}_{+})$\end{document}. We characterize frames, dual frames, and Parseval frames in L2(R+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}(\mathbb {R}_{+})$\end{document} generated by such systems. Interestingly, it turns out that, for such systems, Parseval frames, orthonormal bases, and orthonormal systems are mutually equivalent to each other, while this is not the case for affine systems and Gabor systems.