Advertisement

Knit Product of Finite Groups and Sampling

  • Antonio G. García
  • Miguel A. Hernández-MedinaEmail author
  • Alberto Ibort
Article
  • 25 Downloads

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.

Keywords

Knit product of groups unitary representation of a group finite unitary-invariant subspaces finite frames dual frames left-inverses sampling expansions 

Mathematics Subject Classification

20C40 42C15 94A20 

Notes

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).

References

  1. 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)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brin, M.G.: On the Zappa-Szép product. Commun. Algebra 33(2), 393–424 (2005)CrossRefGoogle Scholar
  3. 3.
    Casazza, P.G., Kutyniok, G. (eds.): Finite Frames: Theory and Applications. Birkhäuser, Boston (2014)Google Scholar
  4. 4.
    Christensen, O.: An Introduction to Frames and Riesz Bases, 2nd edn. Birkhäuser, Boston (2016)zbMATHGoogle Scholar
  5. 5.
    Dodson, M.M.: Groups and the sampling theorem. Sampl. Theory Signal Image Process. 6(1), 1–27 (2007)MathSciNetzbMATHGoogle Scholar
  6. 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)MathSciNetCrossRefGoogle Scholar
  7. 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)MathSciNetCrossRefGoogle Scholar
  8. 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)Google Scholar
  9. 9.
    García, A.G.: Orthogonal sampling formulas: a unified approach. SIAM Rev. 42, 499–512 (2000)MathSciNetCrossRefGoogle Scholar
  10. 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)MathSciNetCrossRefGoogle Scholar
  11. 11.
    García, A.G., Muñoz-Bouzo, M.J.: Sampling-related frames in finite \(U\)-invariant subspaces. Appl. Comput. Harmon. Anal. 39, 173–184 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kluvánek, I.: Sampling theorem in abstract harmonic analysis. Mat.-Fyz. Casopis Sloven. Akad. Vied. 15, 43–48 (1965)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kettle, S.F.A.: Symmetry and Structure: Readable Group Theory for Chemists, 3rd edn. Wiley, New York (2007)Google Scholar
  14. 14.
    Kolmogorov, A.N.: Stationary sequences in Hilbert space. Boll. Moskow. Gos. Univ. Mat. 2, 1–40 (1941)MathSciNetGoogle Scholar
  15. 15.
    Lancaster, P., Tismenetsky, M.: The Theory of Matrices, 2nd edn. Academic Press, Boston (1985)zbMATHGoogle Scholar
  16. 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)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sinha, V.P.: Symmetries and groups in signal processing. Springer, New York (2010)CrossRefGoogle Scholar
  18. 18.
    Stallings, W.T., Boullion, T.L.: The pseudoinverse of an \(r\)-circulant matrix. Proc. Am. Math. Soc. 34, 385–388 (1972)MathSciNetzbMATHGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 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)Google Scholar
  21. 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)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Terras, A.: Fourier Analysis on Finite Groups and Application. LMS Student Texts, vol. 43. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  23. 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)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Carlos III de MadridLeganés-MadridSpain
  2. 2.Departamento de Matemática Aplicada a las Tecnologías de la Información y las Comunicacion, E.T.S.I.T.Information Processing and Telecommunications Center Universidad Politécnica de MadridMadridSpain

Personalised recommendations