Russian Mathematics

, Volume 63, Issue 9, pp 63–83 | Cite as

Comparison of V- and S-Dini Tests. Counterexamples for Symmetric Dini Tests with Respect to Generalized Haar and Walsh Systems

  • V. I. ShcherbakovEmail author


In this paper, we prove that in symmetric Dini tests with respect to Price and generalized Haar systems both conditions are essential; no one of them alone guarantees the convergence of the Fourier series independently of the majorant of Dirichlet kernels. We give the corresponding counterexamples.

Key words

Abelian group Vilenkin group character systems Price systems generalized Haar systems Dirichlet kernels Dini test 


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The author is grateful to Prof. T.P. Lukashenko for valuable remarks.


  1. 1.
    Vilenkin, N.Ya. “A Class of Complete Orthonormal Systems”, Izv. Akad. Nauk SSSR, Ser. Matem. 11 (4), 363–400 (1947).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Agaev, G.N., Vilenkin, N.Ya., Jafarli, G.M., Rubinstein, A.I. Multiplicative Function Systems and Harmonic Analysis on Groups with Measure Zero (ELM, Baku, 1981) [in Russian].Google Scholar
  3. 3.
    Monna, J.L. Analysys non-Archimedence (Springer-Verlag, Berlin, Heidelberg, New York, 1970).CrossRefGoogle Scholar
  4. 4.
    Khrennikov, A.Yu., Shelkovich, V.M. Modern p-Adic Analysis and Mathematical Physics. Theory and Applications (Fizmatgiz, Moscow, 2012) [in Russian].Google Scholar
  5. 5.
    Shcherbakov, V.I. “Pointwise Convergence of Fourier Series with Respect to Multiplicative Systems”, Vestn. MGU Ser.: Matem., Mekh. 2, 37–42 (1983).MathSciNetzbMATHGoogle Scholar
  6. 6.
    Shcherbakov, V.I. “Majorants of the Dirichlet Kernels and the Dini Pointwise Tests for Generalized Haar Systems”, Matem. Zametki 101 (3), 446–473 (2017).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Shcherbakov, V.I. “Dini-Lipschitz on the Generalized Haar Systems”, Izv. Saratovsk. un-ta, Ser.: Matem., Mekh., Inform. 16 (4), 435–448 (2016).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Golubov, B.I. “A Class of Complete Orthogonal Systems”, Sib. Matem. Zhurn. 9 (2), 297–314 (1968).MathSciNetzbMATHGoogle Scholar
  9. 9.
    Shcherbakov, V.I. “Divergence of the Fourier series by Generalized Haar Systems at Points of Continuity of a Function”, Russian Math. (Iz. VUZ) 60 (1), 42–59 (2016).MathSciNetCrossRefGoogle Scholar
  10. 10.
    Price, J.J. “Certain Groups of Orthonormal Step Functions”, Canad. J. Math. 9 (3), 417–425 (1957).MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chrestenson, H.E. “A Class of Generalized Walsh’s Functions”, Pacific J. Math. 5 (1), 17–31 (1955).MathSciNetCrossRefGoogle Scholar
  12. 12.
    Walsh, J.L. “A Closed Set of Normal Orthogonal Functions”, Amer. J. Math. 45 (1), 5–24 (1923).MathSciNetCrossRefGoogle Scholar
  13. 13.
    Paley, R.E.A.C. “A Remarkable Series of Orthonormal Functions”, Proc. London Math. Soc. 36, 241–264 (1932).CrossRefGoogle Scholar
  14. 14.
    Rademacher, H. “Enige Sätze über Reihen von allgemeinen Orthogonalfunctionen”, Math. Ann. 87 (1–2), 112–130 (1922).MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kaczmashz, S., Steinhaus, H. Theory of Orthogonal Series, additions by N.Ya. Vilenkin, §1, item 6, pp. 475–479 (Fizmatgiz, Moscow, 958) [in Russian].Google Scholar
  16. 16.
    Kaczmashz, S., Steinhaus H. Theorie des Orthogonalreichen (Warszawa, Wroclav, 1936).Google Scholar
  17. 17.
    Golubov, B.I., Rubinshtein, A.I. “A Class of Convergence Systems”, Matem. Sborn. 71 (1), 96–115 (1966).MathSciNetGoogle Scholar
  18. 18.
    Vlasova, E.A. Series in the Generalized Haar Systems (Cand. Sci. (Phys.-Math.) dissertation, Moscow, 1987).Google Scholar
  19. 19.
    Vlasova, E.A. “Convergence of Series with Respect to Generalized Haar Systems”, Anal. Math. 13 (4), 339–360 (1987).MathSciNetCrossRefGoogle Scholar
  20. 20.
    Haar, A. “Zur Theorie der Orthogonalischen Functionsysteme”, Math. Ann. 69, 331–371 (1910).MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lukomskii, S.F. “On Haar Series on Compact Zero-Dimensional Groups”, Izv. Saratovsk. Un-ta. Ser.: Matem., Mekh., Inform. 9 (1), 24–29 (2009).MathSciNetGoogle Scholar
  22. 22.
    Komissarova, N.E. “Lebesgue Functions for Haar System on Compact Zero-Dimensional Group”, Izv. Saratovsk. Un-ta. Ser.: Matem., Mekh., Inform. 12 (3), 30–36 (2012).zbMATHGoogle Scholar
  23. 23.
    Berdnikov, G.S. “Graphs with Contours in Multiresolution Analysis on Vilenkin Groups”, Izv. Saratovsk. un-ta, Ser.: Matem., Mekh., Inform. 16 (4), 377–388 (2016).MathSciNetzbMATHGoogle Scholar
  24. 24.
    Fine, N.J. “On the Walsh Function”, Trans. Amer. Math. Soc. 69 (3), 372–414 (1949).MathSciNetCrossRefGoogle Scholar

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© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Moscow Technical University of Communications and InformaticsMoscowRussia

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