# Shift invariant spaces in L2(R,Cm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepa ...

Shift invariant spaces in L2(R,Cm)\documentclass[12pt]{minimal} \usepackage{amsmath}... The paper deals with sampling and reconstruction of vector valued functions in a shift invariant space with multiple generators. Unlike the case of a shift invariant space with multiple generators 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}, when the dimension of the vectors is the same as the number of generators, Z\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {Z}$$\end{document} turns out to be a stable set of sampling. A sampling formula for reconstructing a function from its samples at integer points is derived and the problem of sampling on a perturbed set of integers is discussed. An illustration of sampling and reconstruction of a function in L2(R,R2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{2}(\mathbb {R},\mathbb {R}^{2})$$\end{document} on a finite interval is given using Matlab. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Analysis Springer Journals

# Shift invariant spaces in L2(R,Cm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepa ...

, Volume 29 (2) – Jan 13, 2020
23 pages

/lp/springer-journals/shift-invariant-spaces-in-l2-r-cm-documentclass-12pt-minimal-3vddMkKtI0
Publisher
Springer Journals
ISSN
0971-3611
eISSN
2367-2501
DOI
10.1007/s41478-019-00219-8
Publisher site
See Article on Publisher Site

### Abstract

The paper deals with sampling and reconstruction of vector valued functions in a shift invariant space with multiple generators. Unlike the case of a shift invariant space with multiple generators 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}, when the dimension of the vectors is the same as the number of generators, Z\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {Z}$$\end{document} turns out to be a stable set of sampling. A sampling formula for reconstructing a function from its samples at integer points is derived and the problem of sampling on a perturbed set of integers is discussed. An illustration of sampling and reconstruction of a function in L2(R,R2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^{2}(\mathbb {R},\mathbb {R}^{2})$$\end{document} on a finite interval is given using Matlab.

### Journal

The Journal of AnalysisSpringer Journals

Published: Jan 13, 2020

Keywords: Block Laurent operator; Reproducing kernel Hilbert space; Stable set of sampling; Vector valued Zak transform; 42C15; 94A20

### References

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