Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Smooth orthonormal basis for L2(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2(\mathbb {R}^2)$$\end{document} using smooth projections on L2(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2(\mathbb {R}^2)$$\end{document} associated with rectangles in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document}

Smooth orthonormal basis for L2(R2)\documentclass[12pt]{minimal} \usepackage{amsmath}... For L2(R2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_2(\mathbb {R}^2)$$\end{document}, one can easily construct an orthonormal basis from an orthonormal basis of 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} using the tensor product; and that orthonormal basis is infact separable orthonormal basis. The separable basis have a number of disadvantages, as they have very little design freedom. Furthermore, the separability imposes an unnecessary product structure on R2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^2$$\end{document}, that is artificial for natural images. In this paper, we construct a compactly supported nonseparable smooth orthonormal basis for L2(R2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_2({\mathbb {R}}^2)$$\end{document} by constructing smooth projection PI×M\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_{I\times M}$$\end{document} associated with a rectangle I×M=[a,b]×[c,d]⊂R2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$I\times M=[a,b]\times [c,d]\subset \mathbb {R}^2$$\end{document} on L2(R2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_2({\mathbb {R}}^2)$$\end{document} without using tensor product. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Analysis Springer Journals

Smooth orthonormal basis for L2(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2(\mathbb {R}^2)$$\end{document} using smooth projections on L2(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2(\mathbb {R}^2)$$\end{document} associated with rectangles in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document}

Murugan, S. Pitchai; Youvaraj, G. P.
The Journal of Analysis , Volume OnlineFirst – Sep 22, 2022
Download PDF
23 pages

Loading next page...
 
/lp/springer-journals/smooth-orthonormal-basis-for-l2-r2-documentclass-12pt-minimal-683H6a9W6C
Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to The Forum D’Analystes 2022. Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
0971-3611
eISSN
2367-2501
DOI
10.1007/s41478-022-00488-w
Publisher site
See Article on Publisher Site

Abstract

For L2(R2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_2(\mathbb {R}^2)$$\end{document}, one can easily construct an orthonormal basis from an orthonormal basis of 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} using the tensor product; and that orthonormal basis is infact separable orthonormal basis. The separable basis have a number of disadvantages, as they have very little design freedom. Furthermore, the separability imposes an unnecessary product structure on R2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^2$$\end{document}, that is artificial for natural images. In this paper, we construct a compactly supported nonseparable smooth orthonormal basis for L2(R2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_2({\mathbb {R}}^2)$$\end{document} by constructing smooth projection PI×M\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_{I\times M}$$\end{document} associated with a rectangle I×M=[a,b]×[c,d]⊂R2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$I\times M=[a,b]\times [c,d]\subset \mathbb {R}^2$$\end{document} on L2(R2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_2({\mathbb {R}}^2)$$\end{document} without using tensor product.

Journal

The Journal of Analysis – Springer Journals

Published: Sep 22, 2022

Keywords: Smooth projection; Orthogonal direct sum; Smooth orthonormal basis; 42A65; 42B05; 42C30

References

  • Arbitrarily smooth orthogonal nonseparable wavelets in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R} }^2$$\end{document}
    Belogay, E; Wang, Y
  • Smooth projections and the construction of smooth Parseval frames of shearlets
    Bodmann, BG; Labate, D; Pahari, BR
  • Smooth orthogonal projections on sphere
    Bownik, M; Dziedziul, K
  • Smooth orthogonal projections on Riemannian manifold
    Bownik, M; Dziedziul, K; Kamont, A
  • Riesz basis of Coifman and Meyer’s local sine and cosine type
    Chung, M
  • The construction of the Lemarie-Meyer smooth wavelets with respect to a Riesz Basis
    Chung, M
  • Smooth bump functions and the geometry of Banch spaces: A brief survey
    Fry, R; McManus, S
  • Nonseparable two and three-dimensional wavelets
    Kovacevic, J; Vetterli, M
  • Smooth localized orthonormal bases
    Wickerhauser, MV
  • A smooth non-rectangular time-frequency segmentation of L2(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2({\mathbb{R} }^2)$$\end{document}
    Warner, DW; Healy, DM; Rockmore, DN