Acta Mathematica Hungarica

, Volume 159, Issue 1, pp 42–54 | Cite as

\(\mathcal{J}_H\)-Regular Borel measures on locally compact abelian groups

  • L. Klotz
  • J.M. MedinaEmail author


Let G be an LCA group, H a closed subgroup, \(\varGamma\) the dual group of G. In accordance with analogous notions in prediction theory the classes of \(\mathcal{J}_H\)-regular and \(\mathcal{J}_H\)-singular Borel measures on \({\rm \Gamma}\) are defined. A characterization of \(\mathcal{J}_H\)-regular measures is given and a Wold type decomposition is obtained. Relations to the Whittaker–Shannon–Kotel’nikov theorem are discussed.

Key words and phrases

LCA group regular measure \(L^{\alpha}\)-space trigonometric approximation Whittaker–Shannon–Kotel’nikov theorem 

Mathematics Subject Classification

43A15 42A10 43A05 43A25 


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  1. 1.
    Beaty, M.G., Dodson, M.-M.: The Whittaker-Kotel'nikov-Shannon theorem, spectral translates and Plancherel's formula. J. Fourier Anal. Appl. 10, 179–199 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dodson, M.M., Beaty, M.G., Eveson, S.-P.: A converse to Kluvánek's theorem. J. Fourier Anal. Appl. 13, 187–195 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D. L. Cohn, Measure Theory, Birkhäuser (1980)Google Scholar
  4. 4.
    Mbekhta, M., Ezzahraoui, H., Zerouali, E.H.: Wold-type decomposition for some regular operators. J. Math. Ann. Appl. 430, 483–499 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Feldman, J., Greenleaf, F.P.: Existence of Borel transversals in groups. Pacific J. Math. 25, 455–461 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    U. Grenander and M. Rosenblatt, Statistical Analysis of Stationary Time Series, American Mathematical Society (Providence, 2008) (reprint of the 1957 edition)Google Scholar
  7. 7.
    E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. I, Springer (1963)Google Scholar
  8. 8.
    Kluvánek, I.: Sampling theorem in abstract harmonic analysis. Matematicko-Fyzikalny Casopis 15, 43–47 (1965)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Klotz, L., Riedel, M.: Periodic observations of harmonizable symmetric stable processes. Probab. Math. Statist. 25, 289–306 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Lee, A.J.: Sampling theorems for nonstationary random processes. Trans. Amer. Math. Soc. 242, 225–241 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lloyd, S.P.: A sampling theorem for stationary (wide sense) stochastic processes. Trans. Amer. Math. Soc. 92, 1–12 (1959)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Medina, J.M., Klotz, L., Riedel, M.: Density of spaces of trigonometric polynomials with frequencies from a subgroup in \({L}^{\alpha }\) spaces. C. R. Math. Acad. Sci. Paris 356, 586–593 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    S. A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, Cambridge University Press (Cambridge, 1977)Google Scholar
  14. 14.
    Salehi, H.: On interpolation of \(q\)-variate stationary stochastic processes. Pacific J. Math. 28, 183–191 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Salehi, H., Scheidt, J.K.: Interpolation of \(q\)-variate weakly stationary stochastic processes over a locally compact abelian group. J. Multivariate Anal. 2, 307–331 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Weron, A.: Harmonizable stable processes on groups: spectral, ergodic and interpolation properties. Z. Wahrsch. Verw. Gebiete 68, 473–491 (1985)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Fakultät für Mathematik und InformatikUniversität LeipzigLeipzigGermany
  2. 2.Departamento de MatemáticaInst. Argentino de Matemática “A. P. Calderón” – CONICET, and Universidad de Buenos AiresBuenos AiresArgentina

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