Skip to main content

General weighted class of quaternion-valued functions with lacunary series expansions


The purpose of this article is to define a new general weighted class of hyperholomorphic functions, the so called \({\mathbf {B}}^{q}_{\alpha ,\omega }(G)\) Spaces. For this class we obtain characterizations by weighted Bloch \({{\mathcal {B}}}^{\alpha }_{\omega }\) spaces. Moreover, we characterize the hyperholomorphic \({\mathbf {B}}^{q}_{\alpha ,\omega }(G)\) functions by the coefficients of certain lacunary series expansions in Clifford analysis.

This is a preview of subscription content, access via your institution.


  1. Avetsiyan, K.L.: Hardy–Bloch type spaces and lacunary series in the polydisk. Glasg. Math. J. 49(2), 345–356 (2007)

    MathSciNet  Article  Google Scholar 

  2. Bernstein, S.: Harmonic \(Q_p\) spaces. Comput. Methods Funct. Theory 9(1), 285–304 (2009)

    MathSciNet  Article  Google Scholar 

  3. Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis, Research Notes in Mathematics, vol. 76. Pitman Advanced Publishing Program, London (1982)

  4. Dyakonov, K.M.: Weighted Bloch spaces, \(H^p\) and \(BMOA\). J. Lond. Math. Soc. II. Ser. 65(2), 411–417 (2002)

    Article  Google Scholar 

  5. El-Sayed Ahmed, A.: On weighted \(\alpha \)-Besov spaces and \(\alpha \)-Bloch spaces of quaternion-valued functions. Numer. Funct. Anal. Optim. 29(9–10), 1064–1081 (2008)

    MathSciNet  Article  Google Scholar 

  6. El-Sayed Ahmed, A., Kamal, A.: Series expansions some analytic function spaces. J. Comput. Theor. Nanosci. 12(8), 1586–1593 (2015)

    Article  Google Scholar 

  7. El-Sayed Ahmed, A., Kamal, A., Yassen, T.I.: Characterizations for \( Q_{K,\omega }(p, q)\) type functions by series expansions with Hadamard gaps. CUBO Math. J. 01, 81–93 (2014)

    Article  Google Scholar 

  8. El-Sayed Ahmed, A., Asiri, F.: Characterizations of weighted Bloch space by \(Q_{ p, \omega }\)-type spaces of quaternion-valued functions. J. Comput. Theor. Nanosci. 12, 4250–4255 (2015)

    Article  Google Scholar 

  9. Furdui, O.: On a class of lacunary series in BMOA. J. Math. Anal. Appl. 342(2), 773–779 (2008)

    MathSciNet  Article  Google Scholar 

  10. Gürlebeck, K., Sprössig, W.: Quaternionic Analysis and Elliptic Boundary Value Problems. International Series of Numerical Mathematics, vol. 89. Birkhäuser, Basel (1990)

    Book  Google Scholar 

  11. Gürlebeck, K., Kähler, U., Shapiro, M., Tovar, L.M.: On \({\mathbf{Q}_p}\) spaces of quaternion-valued functions. Complex Var. 39, 115–135 (1999)

    MATH  Google Scholar 

  12. Gürlebeck, K., Malonek, H.R.: On strict inclusions of weighted Dirichlet spaces of monogenic functions. Bull. Aust. Math. Soc. 64, 33–50 (2001)

    MathSciNet  Article  Google Scholar 

  13. Gürlebeck, K., El-Sayed Ahmed, A.: Integral norms for hyperholomorphic Bloch-functions in the unit ball of \({\mathbb{R}}^3\). In: Begehr et al. (eds.) Progress in Analysis. Kluwer Academic, Dordrecht, pp. 253–263 (2003)

  14. Gürlebeck, K., El-Sayed Ahmed, A.: On series expansions of hyperholomorphic \(B^q\) functions. Trends Math. Adv. Anal. Geom. 113–129 (2004)

  15. Gürlebeck, K., El-Sayed Ahmed, A.: On \(B^q\) spaces of hyperholomorphic functions and the Bloch space in \(R^3\). In: Finite or Infinity Dimensional, Complex Analysis and Its Applications, pp. 269–286 (2004)

  16. Kamal, A., Ahmed, A., Yassen, T.I.: Quasi-metric spaces and composition operators on \( {\cal{B}}^{*}_{\alpha , \log }\) and \(Q*_{ p, \log }\) spaces. J. Comput. Theor. Nanosci. 2(8), 1795–1801 (2015)

    Article  Google Scholar 

  17. Kamal, A., Yassen, T.I.: Existence solutions of the complex linear differential equations in \(Q_{K, \omega } (p, q) \) spaces. Gen. Lett. Math. 3(1), 47–56 (2017)

    Google Scholar 

  18. Kamal, A., Yassen, T.I.: A property of hyperbolic general family of function spaces with Hadamard gap series. J. Adv. St. Topol. 1–9 (2018)

  19. Li, S., Stević, S.: Weighted-Hardy functions with Hadamard gaps on the unit ball. Appl. Math. Comput. 212(1), 229–233 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Meng, X.: Some sufficient conditions for analytic functions to belong to \(Q_{K,0}(p, q)\) space1. Abstr. Appl. Anal. 2008, 404636 (2008)

    Article  Google Scholar 

  21. Malonek, M.: Power series representation for monogenic functions in \(R^{m+1}\) based on a permutational product. Complex Var. Theory Appl. 15, 181–191 (1990)

    MathSciNet  MATH  Google Scholar 

  22. Miao, J.: A property of analytic functions with Hadamard gaps. Bull. Aust. Math. Soc. 45, 105–112 (1992)

    MathSciNet  Article  Google Scholar 

  23. Rashwan, R.A., El-Sayed Ahmed, A., Kamal, A.: Integral characterizations of weighted Bloch spaces and \(Q_{K,\omega } (p,q)\) spaces. Math. Tome 51(74)(2), 63–76 (2009)

  24. Reséndis, L. F., Tovar, L. M.: Besov-type characterizations for quaternionic Bloch functions. In: Son, L.H., et al. (eds.) Finite or Infinite Complex Analysis and Its Applications. Advances in Complex Analysis and Applications, pp. 207–220. Kluwer Academic Publishers, Boston (2004)

  25. Stević, S.: Bloch-type functions with Hadamard gaps. App. Math. Comput. 208, 416–422 (2009)

    MathSciNet  Article  Google Scholar 

  26. Stroethoff, K.: Besov-type characterizations for the Bloch space. Bull. Aust. Math. Soc. 39, 405–420 (1989)

    MathSciNet  Article  Google Scholar 

  27. Wulan, H., Zhu, K.: Lacunary series in \(Q_K\) spaces. Stud. Math. 178, 217–230 (2007)

    Article  Google Scholar 

Download references


We would like to express our gratitude to the anonymous referees for their helpful suggestions and corrections.The authors declare that they have no competing interests.

Author information

Authors and Affiliations



The authors contributed equally in writing the final version of this article. All authors read and approved the final manuscript. Please refer to Journal-level guidance for any specific requirements.

Corresponding author

Correspondence to Alaa Kamal.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Verify currency and authenticity via CrossMark

Cite this article

Kamal, A., Yassen, T.I. General weighted class of quaternion-valued functions with lacunary series expansions. Afr. Mat. 33, 69 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


  • Quaternionic analysis
  • \({\mathbf {B}}^{q}_{\alpha , \omega }(G)\) spaces
  • Hyperholomorphic functions
  • Clifford analysis

Mathematics Subject Classification

  • 32Axx
  • 32Hxx