Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Pure and Applied Mathematics

Editor-in-Chief: Trimeche, Khalifa

Editorial Board: Aldroubi, Akram / Anker, Jean-Philippe / Aouadi, Saloua / Bahouri, Hajer / Baklouti, Ali / Bakry, Dominique / Baraket, Sami / Ben Abdelghani, Leila / Begehr, Heinrich / Beznea, Lucian / Bezzarga, Mounir / Bonami, Aline / Demailly, Jean-Pierre / Fleckinger, Jacqueline / Gallardo, Leonard / Ismail, Mourad / Jarboui, Noomen / Jouini, Elyes / Karoui, Abderrazek / Kamoun, Lotfi / Kobayashi, Toshiyuki / Maday, Yvon / Marzougui, Habib / Mili, Maher / Mustapha, Sami / Ovsienko, Valentin / Peigné, Marc / Pouzet, Maurice / Radulescu, Vicentiu / Schwartz, Lionel / Sifi, Mohamed / Zaag, Hatem / Zarati, Said

4 Issues per year


CiteScore 2017: 1.29

SCImago Journal Rank (SJR) 2017: 0.177
Source Normalized Impact per Paper (SNIP) 2017: 0.409

Mathematical Citation Quotient (MCQ) 2017: 0.21

Online
ISSN
1869-6090
See all formats and pricing
More options …

A sampling theorem for the twisted shift-invariant space

Radha Ramakrishnan / Saswata Adhikari
Published Online: 2017-06-17 | DOI: https://doi.org/10.1515/apam-2016-0090

Abstract

Recently, a characterization of frames in twisted shift-invariant spaces in L2(2n) has been obtained in [16]. Using this result, we prove a sampling theorem on a subspace of a twisted shift-invariant space in this paper.

Keywords: Canonical dual frames; frames; sampling theorem; shift-invariant space; twisted translation

MSC 2010: 94A20; 42C15; 42B99

References

  • [1]

    A. E. Acosta, A. Aldroubi and I. Krishtal, On stability of sampling reconstruction models, Adv. Comput. Math. 31 (2008), 5–34. Web of ScienceGoogle Scholar

  • [2]

    A. Aldroubi and K. Gröchenig, Beurling–Landau-type theorems for non-uniform sampling in shift-invariant spline spaces, J. Fourier Anal. Appl. 6 (2000), 93–103. CrossrefGoogle Scholar

  • [3]

    A. Aldroubi and K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Rev. 43 (2001), no. 4, 585–620. CrossrefGoogle Scholar

  • [4]

    M. Bownik, The structure of shift-invariant subspaces of L2(n), J. Funct. Anal. 176 (2000), 282–309. Google Scholar

  • [5]

    C. Cabrelli and V. Paternostro, Shift-invariant spaces on LCA groups, J. Funct. Anal. 258 (2010), 2034–2059. CrossrefWeb of ScienceGoogle Scholar

  • [6]

    O. Christensen, Frames and Bases. An Introductory Course, Birkhäuser, Boston, 2008. Google Scholar

  • [7]

    R. J. Duffin and J. J. Eachus, Some notes on an expansion theorem of Paley and Wiener, Bull. Amer. Math. Soc. 48 (1942), 850–855. CrossrefGoogle Scholar

  • [8]

    H. G. Feichtinger and K. Gröchenig, Theory and practice of irregular sampling, Wavelets: Mathematics and Applications, Stud. Adv. Math., CRC Press, Boca Raton (1994), 305–363. Google Scholar

  • [9]

    A. G. García, G. Pérez-Villalón and A. Portal, Riesz bases in L2(0,1) related to sampling in shift-invariant spaces, J. Math. Anal. Appl. 308 (2005), 703–713. Google Scholar

  • [10]

    K. Gröchenig and H. Schwab, Fast local reconstruction methods for nonuniform sampling in shift-invariant spaces, SIAM J. Matrix Anal. Appl. 24 (2003), 899–913. CrossrefGoogle Scholar

  • [11]

    K. Gröchenig and J. Stöckler, Gabor frames and totally positive functions, Duke Math. J. 162 (2013), 1003–1031. Web of ScienceCrossrefGoogle Scholar

  • [12]

    C. Heil, A Basis Theory Primer. Expanded Edition, Birkhäuser, Basel, 2011. Web of ScienceGoogle Scholar

  • [13]

    M. I. Kadec, The exact value of the Paley–Wiener constant, Dokl. Akad. Nauk. SSSR. 155 (1964), 1253–1254. Google Scholar

  • [14]

    R. A. Kamyabi Gol and R. Raisi Tousi, The structure of shift invariant spaces on a locally compact abelian group, J. Math. Anal. Appl. 340 (2008), 219–225. CrossrefWeb of ScienceGoogle Scholar

  • [15]

    R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, Amer. Math. Soc. Colloq. Publ. 19, American Mathematical Society, Providence, 1987. Google Scholar

  • [16]

    R. Radha and S. Adhikari, Frames and Riesz bases of twisted shift-invariant spaces in L2(2n), J. Math. Anal. Appl. 434 (2016), 1442–1461. Google Scholar

  • [17]

    R. Radha and N. Shravan Kumar, Shift-invariant subspaces on compact groups, Bull. Sci. Math. 137 (2013), no. 4, 485–497. CrossrefGoogle Scholar

  • [18]

    Q. Sun, Local reconstruction for sampling in shift-invariant spaces, Adv. Comput. Math. 32 (2010), 335–352. Web of ScienceCrossrefGoogle Scholar

  • [19]

    W. Sun and X. Zhou, Average sampling in spline subspaces, Appl. Math. Lett. 15 (2002), 233–237. CrossrefGoogle Scholar

  • [20]

    W. Sun and X. Zhou, Reconstruction of functions in spline subspaces from local averages, Proc. Amer. Math. Soc. 131 (2003), 2561–2571. CrossrefGoogle Scholar

  • [21]

    S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Birkhäuser, Boston, 1997. Google Scholar

  • [22]

    G. G. Walter and X. Shen, Wavelets and Other Orthogonal Systems, 2nd ed., Stud. Adv. Math., Chapman and Hall/CRC, Boca Raton, 2001. Google Scholar

  • [23]

    X. Zhou and W. Sun, On the sampling theorem for wavelet subspaces, J. Fourier Anal. Appl. 5 (1999), no. 4, 347–354. CrossrefGoogle Scholar

About the article

Received: 2016-09-09

Revised: 2017-04-27

Accepted: 2017-05-01

Published Online: 2017-06-17

Published in Print: 2017-10-01


Citation Information: Advances in Pure and Applied Mathematics, Volume 8, Issue 4, Pages 293–305, ISSN (Online) 1869-6090, ISSN (Print) 1867-1152, DOI: https://doi.org/10.1515/apam-2016-0090.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in