Subspace dual and orthogonal frames by action of an abelian group

被引:2
作者
Sarkar, Sudipta [1 ]
Shukla, Niraj K. [1 ]
机构
[1] Indian Inst Technol Indore, Dept Math, Khandwa Rd, Indore 453552, India
关键词
Locally compact group; Zak transform; Translation invariant system; Subspace dual and orthogonal frames; Biorthogonal system; INVARIANT-SYSTEMS; GABOR FRAMES; TRANSLATION; REPRESENTATIONS; SPACES; PAIR;
D O I
10.1007/s11868-024-00594-2
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
In this article, we discuss subspace duals of a frame of translates by an action of a closed abelian subgroup Gamma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} of a locally compact group G.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {G}}.$$\end{document} These subspace duals are not required to lie in the space generated by the frame. We characterise translation-generated subspace duals of a frame/Riesz basis involving the Zak transform for the pair (G,Gamma).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathscr {G}}, \Gamma ).$$\end{document} We continue our discussion on the orthogonality of two translation-generated Bessel pairs using the Zak transform, which allows us to explore the dual of super-frames. As an example, we extend our findings to splines, Gabor systems, p-adic fields Qp,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Q}} p,$$\end{document} locally compact abelian groups using the fiberization map.
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收藏
页数:33
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