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

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Abstract

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.

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References

  1. Vilenkin, N.Ya. “A Class of Complete Orthonormal Systems”, Izv. Akad. Nauk SSSR, Ser. Matem. 11 (4), 363–400 (1947).

    MathSciNet  MATH  Google Scholar 

  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. Monna, J.L. Analysys non-Archimedence (Springer-Verlag, Berlin, Heidelberg, New York, 1970).

    Book  Google Scholar 

  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. Shcherbakov, V.I. “Pointwise Convergence of Fourier Series with Respect to Multiplicative Systems”, Vestn. MGU Ser.: Matem., Mekh. 2, 37–42 (1983).

    MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  7. Shcherbakov, V.I. “Dini-Lipschitz on the Generalized Haar Systems”, Izv. Saratovsk. un-ta, Ser.: Matem., Mekh., Inform. 16 (4), 435–448 (2016).

    MathSciNet  MATH  Google Scholar 

  8. Golubov, B.I. “A Class of Complete Orthogonal Systems”, Sib. Matem. Zhurn. 9 (2), 297–314 (1968).

    MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  10. Price, J.J. “Certain Groups of Orthonormal Step Functions”, Canad. J. Math. 9 (3), 417–425 (1957).

    MathSciNet  MATH  Google Scholar 

  11. Chrestenson, H.E. “A Class of Generalized Walsh’s Functions”, Pacific J. Math. 5 (1), 17–31 (1955).

    Article  MathSciNet  Google Scholar 

  12. Walsh, J.L. “A Closed Set of Normal Orthogonal Functions”, Amer. J. Math. 45 (1), 5–24 (1923).

    Article  MathSciNet  Google Scholar 

  13. Paley, R.E.A.C. “A Remarkable Series of Orthonormal Functions”, Proc. London Math. Soc. 36, 241–264 (1932).

    Article  Google Scholar 

  14. Rademacher, H. “Enige Sätze über Reihen von allgemeinen Orthogonalfunctionen”, Math. Ann. 87 (1–2), 112–130 (1922).

    Article  MathSciNet  Google Scholar 

  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].

  16. Kaczmashz, S., Steinhaus H. Theorie des Orthogonalreichen (Warszawa, Wroclav, 1936).

    Google Scholar 

  17. Golubov, B.I., Rubinshtein, A.I. “A Class of Convergence Systems”, Matem. Sborn. 71 (1), 96–115 (1966).

    MathSciNet  Google Scholar 

  18. Vlasova, E.A. Series in the Generalized Haar Systems (Cand. Sci. (Phys.-Math.) dissertation, Moscow, 1987).

  19. Vlasova, E.A. “Convergence of Series with Respect to Generalized Haar Systems”, Anal. Math. 13 (4), 339–360 (1987).

    Article  MathSciNet  Google Scholar 

  20. Haar, A. “Zur Theorie der Orthogonalischen Functionsysteme”, Math. Ann. 69, 331–371 (1910).

    Article  MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    MATH  Google Scholar 

  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).

    MathSciNet  MATH  Google Scholar 

  24. Fine, N.J. “On the Walsh Function”, Trans. Amer. Math. Soc. 69 (3), 372–414 (1949).

    Article  MathSciNet  Google Scholar 

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Acknowledgments

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

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Authors and Affiliations

  1. Moscow Technical University of Communications and Informatics, 32 Narodnogo opolcheniya str., Moscow, 123995, Russia

    V. I. Shcherbakov

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  1. V. I. Shcherbakov
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Correspondence to V. I. Shcherbakov.

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Russian Text © The Author(s), 2019, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2019, No. 9, pp. 73–95.

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Shcherbakov, V.I. Comparison of V- and S-Dini Tests. Counterexamples for Symmetric Dini Tests with Respect to Generalized Haar and Walsh Systems. Russ Math. 63, 63–83 (2019). https://doi.org/10.3103/S1066369X1909007X

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  • DOI: https://doi.org/10.3103/S1066369X1909007X

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