[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content

Knit Product of Finite Groups and Sampling

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

A finite sampling theory associated with a unitary representation of a finite non-abelian group \({\mathbf {G}}\) on a Hilbert space is established. The non-abelian group \({\mathbf {G}}\) is a knit product \({\mathbf {N}}\bowtie {\mathbf {H}}\) of two finite subgroups \({\mathbf {N}}\) and \({\mathbf {H}}\) where at least \({\mathbf {N}}\) or \({\mathbf {H}}\) is abelian. Sampling formulas where the samples are indexed by either \({\mathbf {N}}\) or \({\mathbf {H}}\) are obtained. Using suitable expressions for the involved samples, the problem is reduced to obtain dual frames in the Hilbert space \(\ell ^2({\mathbf {G}})\) having a unitary invariance property; this is done by using matrix analysis techniques. An example involving dihedral groups illustrates the obtained sampling results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Barbieri, D., Hernández, E., Parcet, J.: Riesz and frame systems generated by unitary actions of discrete groups. Appl. Comput. Harmon. Anal. 39(3), 369–399 (2015)

    Article  MathSciNet  Google Scholar 

  2. Brin, M.G.: On the Zappa-Szép product. Commun. Algebra 33(2), 393–424 (2005)

    Article  Google Scholar 

  3. Casazza, P.G., Kutyniok, G. (eds.): Finite Frames: Theory and Applications. Birkhäuser, Boston (2014)

    Google Scholar 

  4. Christensen, O.: An Introduction to Frames and Riesz Bases, 2nd edn. Birkhäuser, Boston (2016)

    MATH  Google Scholar 

  5. Dodson, M.M.: Groups and the sampling theorem. Sampl. Theory Signal Image Process. 6(1), 1–27 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Fernández-Morales, H.R., García, A.G., Hernández-Medina, M.A., Muñoz-Bouzo, M.J.: Generalized sampling: from shift-invariant to \(U\)-invariant spaces. Anal. Appl. 13(3), 303–329 (2015)

    Article  MathSciNet  Google Scholar 

  7. Fernández-Morales, H.R., García, A.G., Muñoz-Bouzo, M.J., Ortega, A.: Finite sampling in multiple generated \(U\)-invariant subspaces. IEEE Trans. Inf. Theory 62(4), 2203–2212 (2016)

    Article  MathSciNet  Google Scholar 

  8. Frazier, M.W., Torres, R.: The sampling theorem, \(\varphi \)-transform, and Shannon wavelets for \({\mathbb{R}}\), \({\mathbb{Z}}\), \({\mathbb{T}}\), and \({\mathbb{Z}}_N\). In: Benedetto, J.J., Frazier, M.W. (eds.) Wavelets. Mathematics and Applications, pp. 221–245. CRC Press, Boca Raton FL (1994)

  9. García, A.G.: Orthogonal sampling formulas: a unified approach. SIAM Rev. 42, 499–512 (2000)

    Article  MathSciNet  Google Scholar 

  10. García, A.G., Pérez-Villalón, G.: Dual frames in \({L}^2(0,1)\) connected with generalized sampling in shift-invariant spaces. Appl. Comput. Harmon. Anal. 20(3), 422–433 (2006)

    Article  MathSciNet  Google Scholar 

  11. García, A.G., Muñoz-Bouzo, M.J.: Sampling-related frames in finite \(U\)-invariant subspaces. Appl. Comput. Harmon. Anal. 39, 173–184 (2015)

    Article  MathSciNet  Google Scholar 

  12. Kluvánek, I.: Sampling theorem in abstract harmonic analysis. Mat.-Fyz. Casopis Sloven. Akad. Vied. 15, 43–48 (1965)

    MathSciNet  MATH  Google Scholar 

  13. Kettle, S.F.A.: Symmetry and Structure: Readable Group Theory for Chemists, 3rd edn. Wiley, New York (2007)

    Google Scholar 

  14. Kolmogorov, A.N.: Stationary sequences in Hilbert space. Boll. Moskow. Gos. Univ. Mat. 2, 1–40 (1941)

    MathSciNet  Google Scholar 

  15. Lancaster, P., Tismenetsky, M.: The Theory of Matrices, 2nd edn. Academic Press, Boston (1985)

    MATH  Google Scholar 

  16. Pye, W.C., Boullion, T.L., Atchison, T.A.: The pseudoinverse of a composite matrix of circulants. SIAM J. Appl. Math. 24, 552–555 (1973)

    Article  MathSciNet  Google Scholar 

  17. Sinha, V.P.: Symmetries and groups in signal processing. Springer, New York (2010)

    Book  Google Scholar 

  18. Stallings, W.T., Boullion, T.L.: The pseudoinverse of an \(r\)-circulant matrix. Proc. Am. Math. Soc. 34, 385–388 (1972)

    MathSciNet  MATH  Google Scholar 

  19. Stankovic, R.S., Moraga, C., Astola, J.T.: Fourier Analysis on Finite Groups with Applications in Signal Processing and System Design. Wiley-Interscience, New Jersey (2005)

    Book  Google Scholar 

  20. Stankovic, R.S., Astola, J.T., Karpovsky, M.G.: Some historical remarks on sampling theorem. In: Proceedings of the 2006 International TICSP Workshop on Spectral Methods and Multirate Signal Processing, SMMSP2006, Florence, Italy (2006)

  21. Szép, J.: On the structure of groups which can be represented as the product of two subgroups. Acta Sci. Math. Szeged 12, 57–61 (1950)

    MathSciNet  MATH  Google Scholar 

  22. Terras, A.: Fourier Analysis on Finite Groups and Application. LMS Student Texts, vol. 43. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  23. Zappa, G.: Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili traloro, pp. 119–125. Atti Secondo Congresso Un. Mat. Ital., Bologna; Edizioni Cremonense, Rome (1942)

Download references

Acknowledgements

The authors wish to thank the referee for his/her valuable and constructive comments. This work has been supported by the Grant MTM2017-84098-P from the Spanish Ministerio de Economía y Competitividad (MINECO).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Miguel A. Hernández-Medina.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

García, A.G., Hernández-Medina, M.A. & Ibort, A. Knit Product of Finite Groups and Sampling. Mediterr. J. Math. 16, 146 (2019). https://doi.org/10.1007/s00009-019-1417-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-019-1417-8

Keywords

Mathematics Subject Classification