Results in Mathematics

, 74:34 | Cite as

A Completeness Theorem for a Hahn–Fourier System and an Associated Classical Sampling Theorem

  • H. A. HassanEmail author


We introduce a completeness theorem for a Hahn–Fourier-type trigonometric system, where the sine and cosine functions are replaced by \(q,\omega \)-trigonometric functions, where \(0<q<1,0<\omega \) are fixed. The completeness is established in an appropriate \(L^2\)-space, defined in terms of Jackson–Nörlund integrals. We then derive a \(q,\omega \)-counterpart of the celebrated sampling theorem of Whittaker (Proc R Soc Edinb 35:181–194, 1915), as reported by Kotel’nikov (in: Material for the first all union conference on questions of communications, Moscow, 1933) and Shannon (Proc IRE 37:10–21, 1949), for finite Hahn–Fourier-type integral transforms. A convergence analysis is established and comparative numerical examples are exhibited.


Hahn difference operator Jackson–Nörlund transform sampling theory 

Mathematics Subject Classification

39A10 44A55 94A20 



The author thank the anonymous referee for his careful reading of the paper and for the constructive comments.


  1. 1.
    Abreu, L.D.: A \(q\)-sampling theorem related to the \(q\)-Hankel transform. Proc. Am. Math. Soc. 133, 1197–1203 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abreu, L.D.: Sampling theory associated with q-difference equations of the Sturm–Liouville type. J. Phys. A: Math. Gen. 38, 10311–10319 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Abreu, L.D.: Completeness, special functions and uncertainty principles over q-linear grids. J. Phys. A: Math. Gen. 39, 14567–14580 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Abreu, L.D., Bustoz, J.: On the completeness of sets of \(q\)-Bessel functions \(J^{(3)}_\nu (x;q)\). In: Ismail, M.E.H., Koelink, E. (eds.) Theory and Applications of Special Functions, pp. 29–38. Springer, Boston (2005)CrossRefGoogle Scholar
  5. 5.
    Annaby, M.H.: \(q\)-type sampling theorems. Results Math. 206, 214–225 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Annaby, M.H., Bustoz, J., Ismail, M.E.H.: On sampling theory and basic Sturm–Liouville systems. J. Comput. Appl. Math. 206, 73–85 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Annaby, M.H., Hamza, A.E., Aldwoah, K.A.: Hahn difference operator and associated Jackson–Nörlund integrals. J. Optim. Theory Appl. 154, 133–153 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Annaby, M.H., Hamza, A.E., Makharesh, S.D.: A Sturm–Liouville theory for Hahn difference operator. In: Li, X., Nashed, Z. (eds.) Frontiers of Orthogonal Polynomials and \(q\)-Series, pp. 35–84. World Scientifc, Singapore (2018)CrossRefGoogle Scholar
  9. 9.
    Annaby, M.H., Hassan, H.A., Mansour, Z.S.: Sampling theorems associated with singular \(q\)-Sturm Liouville problems. Results. Math. 62, 121–136 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Annaby, M.H., Hassan, H.A.: Sampling theorems for Jackson–Nörlund transforms associated with Hahn-difference operators. J. Math. Anal. Appl. 464, 493–506 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Annaby, M.H., Mansour, Z.S.: On the zeros of the second and third Jackson q-Bessel functions and their associated q-Hankel transforms. Math. Proc. Camb. Phil. Soc. 147, 47–67 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Annaby, M.H., Mansour, Z.S.: Asymptotic formulae for eigenvalues and eigenfunctions of q-Sturm–Liouville problems. Math. Nachr. 284, 443–470 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Annaby, M.H., Mansour, Z.S.: \(q\)-Fractional Calculus and Equations. Springer, Berlin (2012)CrossRefGoogle Scholar
  14. 14.
    Bustoz, J., Cardoso, J.L.: Basic analog of Fourier series on a q-linear grid. J. Approx. Theory 112, 134–157 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Cardoso, J.L.: Basic Fourier series: convergence on and outside the q-Linear grid. J. Fourier Anal. Appl. 17, 96–114 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cardoso, J.L., Petronilho, J.: Variations around Jackson’s quantum operator. Methods Appl. Anal. 22, 343–358 (2015)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Hahn, W.: Über orthogonalpolynome, die q-differenzenlgleichungen genügen. Math. Nachr. 2, 4–34 (1949)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hira, F.: Sampling theorem associated with \(q\)-Dirac system, preprint. arXiv:1804.06224v1
  19. 19.
    Ismail, M.E.H., Zayed, A.I.: A \(q\)-analogue of the Whittaker–Shannon–Kotel’nikov sampling theorem. Proc. Am. Math. Soc. 131, 3711–3719 (2003)CrossRefGoogle Scholar
  20. 20.
    Kotel’nikov, V.: On the carrying capacity of the ether and wire in telecommunications. In: Material for the first all union conference on questions of communications, (Russian) Izd. Red. Upr. Svyazi RKKA, Moscow (1933)Google Scholar
  21. 21.
    Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1989)zbMATHGoogle Scholar
  22. 22.
    Shannon, C.: Communication in the presence of noise. Proc. IRE 37, 10–21 (1949)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Suslov, S.K.: Some expansions in basic Fourier series and related topics. J. Approx. Theory 115, 289–353 (2002)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Whittaker, E.: On the functions which are represented by the expansion of the interpolation theory. Proc. R. Soc. Edinb. (A) 35, 181–194 (1915)CrossRefGoogle Scholar
  25. 25.
    Zayed, A.I.: Advances in Shannons Sampling Theory. CRC Press, Boca Raton (1993)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceCairo UniversityGizaEgypt
  2. 2.Department of Mathematics, Faculty of Basic EducationPAAETAdailiyahKuwait

Personalised recommendations