Skip to main content
Springer Nature Link
Log in

On Divergence of Fejér Means with Respect to Walsh System on Sets of Measure Zero

  • Conference paper
  • First Online:
Tbilisi Analysis and PDE Seminar (TAPDES 2023)

Part of the book series: Trends in Mathematics ((RPGAPC,volume 7))

Included in the following conference series:

  • 47 Accesses

Abstract

In this paper we prove by the concrete construction that for any set E of measure zero there exists a function \(f\in L_p(G) (1\leq p<\infty )\) such that the Féjer means with respect to Wals system diverge on this set. The key is to use new constructions of Walsh polynomials, which was introduced in Persson et al. (Martingale Hardy Spaces and Summability of One-dimensional Vilenkin-Fourier Series. Birkhäuser/Springer, 2022). In fact, the theorem we prove follows from the general result of Karagulyan (Mat. Sb. 202(1):11–36, 2011), but we provide an alternative approach and the constructed function in our proof has a simpler explicit representation.

The research was supported by Shota Rustaveli National Science Foundation grant no. MR-23-173.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
eBook
USD 189.00
Price excludes VAT (USA)
Hardcover Book
USD 219.99
Price excludes VAT (USA)

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. G.N. Agaev, N.Y. Vilenkin, G.M. Dzhafarly, A.I. Rubinshtein, Multiplicative Systems of Functions and Harmonic Analysis on Zero-dimensional Groups (Elm, Baku, 1981) (Russian)

    Google Scholar 

  2. D. Baramidze, K. Nagy, L.E. Persson, G. Tephnadze, A sharp boundedness result concerning maximal operators of Vilenkin-Fourier series on martingale Hardy spaces. Georgian Math. J. 26(3), 351–360 (2019)

    Article  MathSciNet  Google Scholar 

  3. D. Baramidze, N. Nadirashvili, L.-E. Persson, G. Tephnadze, Some weak-type inequalities and almost everywhere convergence of Vilenkin-Nörlund means. J. Inequal. Appl. 2023, paper no. 66, 17 pp. (2023)

    Google Scholar 

  4. P. Billard, Sur la convergence presque partout des séries de Fourier-Walsh des fonctions de l’espace \(L^2 [0,1]\). Stud. Math. 28, 363–388 (1966/1967)

    Google Scholar 

  5. K. Bitsadze, On divergence of Fourier series with respect to multiplicative systems on the sets of measure zero. Georgian Math. J. 16(3), 435–448 (2009)

    Article  MathSciNet  Google Scholar 

  6. V.M. Bugadze, Divergence of Fourier-Walsh series of bounded functions on sets of measure zero. Mat. Sb. 185(7), 119–127 (1994) (in Russian); English transl.: Russian Math. Sb. 185, 365–372 (1994)

    Google Scholar 

  7. V.M. Bugadze, On divergence of Fourier-Walsh series of bounded functions on sets of measure zero, Russian Academy of Sciences. Sb. Math. 82 (2), 365–372 (1995)

    MathSciNet  Google Scholar 

  8. V.V. Buzdalin, Unboundedly diverging trigonometric Fourier series of continuous functions. Mat. Zametki 7(1), 7–18 (1970) (in Russian); English transl.: Math. Notes 7(1), 5–12 (1970)

    Google Scholar 

  9. L. Carleson, On convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157 (1966)

    Article  MathSciNet  Google Scholar 

  10. C. Demeter, A guide to Carleson’s theorem. Rocky Mountain J. Math. 45(1), 169–212 (2015)

    Article  MathSciNet  Google Scholar 

  11. Ch. Fefferman, Pointwise convergence of Fourier series. Ann. Math. 98(3), 551–571 (1973)

    Article  MathSciNet  Google Scholar 

  12. D.H. Fremlin, Measure Theory, vol. 2, Broad Foundations. Corrected second printing of the 2001 original (Torres Fremlin, Colchester, 2003)

    Google Scholar 

  13. U. Goginava, On divergence of Walsh-Fejer means of bounded functions on sets of measure zero. Acta Math. Hungar. 121(3), 359–369 (2008)

    Article  MathSciNet  Google Scholar 

  14. L. Grafakos, Classical Fourier Analysis, 3rd edn, vol. 249. Graduate Texts in Mathematics (Springer, New York, 2014)

    Google Scholar 

  15. R. Hunt, On the convergence of Fourier series, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967) (Southern Illinois Univ. Press, Carbondale, 1968), pp. 235–255

    Google Scholar 

  16. J.-P. Kahane, Y. Katznelson, Sur les ensembles de divergence des séries trigonométriques. Stud. Math. 26, 305–306 (1966)

    Article  Google Scholar 

  17. G.A. Karagulyan, Divergence of general operators on sets of measure zero. Colloquium Math. 121(1), 113–119 (2010)

    Article  MathSciNet  Google Scholar 

  18. G.A. Karagulyan, On a characterization of the sets of divergence points of sequences of operators with the localization property (Russian). Mat. Sb., 202(1), 11–36 (2011)

    MathSciNet  Google Scholar 

  19. Sh.V. Kheladze, On the everywhere divergence of Fourier-Walsh series. Soobshch. Akad. Nauk. Gruzin. 77, 305–307 (1975)

    MathSciNet  Google Scholar 

  20. Sh.V. Kheladze, On everywhere divergence of Fourier series with respect to bounded type Vilenkin systems. Trudy Tbiliss. Mat. Inst. Gruzin. SSR (in Russian) 58, 225–242 (1978)

    MathSciNet  Google Scholar 

  21. A.N. Kolmogorov, Une série de Fourier-Lebesgue divergente presque partout. Polska Akad. Nauk. Fund. Math. 4, 324–328 (1923)

    Article  Google Scholar 

  22. M.T. Lacey, Carleson’s theorem: proof, complements, variations. Publ. Mat. 48(2), 251–307 (2004)

    Article  MathSciNet  Google Scholar 

  23. M. Lacey, C. Thiele, A proof of boundedness of the Carleson operator. Math. Res. Lett. 7(4), 361–370 (2000)

    Article  MathSciNet  Google Scholar 

  24. N.N. Luzin, Collected works. In: Metric Theory of Function and Functions of a Complex Variable, vol. I. (Russian) (Izdat. Acad. Nauk. SSSR, Moscow, 1953)

    Google Scholar 

  25. N. Nadirashvili, L.-E. Persson, G. Tephnadze, F. Weisz, Vilenkin-Lebesgue points and almost everywhere convergence for some classical summability methods. Mediterr. J. Math. 19, 239 (2022)

    Article  MathSciNet  Google Scholar 

  26. G.G. Oniani, On the divergence sets of Fourier series in systems of characters of compact abelian groups (Russian); translated from Mat. Zametki 112(1), 95–105 (2022). Math. Notes 112(1–2), 100–108 (2022)

    Google Scholar 

  27. L.-E. Persson, G. Tephnadze, P. Wall, On the maximal operators of Vilenkin-Nörlund means. J. Fourier Anal. Appl. 21(1), 76–94 (2015)

    Article  MathSciNet  Google Scholar 

  28. L.-E. Persson, F. Schipp, G. Tephnadze, F. Weisz, An analogy of the Carleson-Hunt theorem with respect to Vilenkin systems. J. Fourier Anal. Appl. 28(48), 1–29 (2022)

    MathSciNet  Google Scholar 

  29. L.-E. Persson, G. Tephnadze, F. Weisz, Martingale Hardy Spaces and Summability of One-dimensional Vilenkin-Fourier Series (Birkhäuser/Springer, 2022)

    Google Scholar 

  30. F. Schipp, Über die divergenz der Walsh-Fourierreihen. Ann. Univ. Sci. Budapest Eötvös Sect. Math. 12, 49–62 (1969)

    MathSciNet  Google Scholar 

  31. F. Schipp, Certain rearrangements of series in the Walsh series. Mat. Zametki 18, 193–201 (1975)

    MathSciNet  Google Scholar 

  32. F. Schipp, Pointwise convergence of expansions with respect to certain product systems. Anal. Math. 2(1), 65–76 (1976)

    Article  MathSciNet  Google Scholar 

  33. F. Schipp, W.R. Wade, P. Simon, J. Pál, Walsh series. An Introduction to Dyadic Harmonic Analysis (Adam Hilger, Bristol, 1990)

    Google Scholar 

  34. P. Sjölin, An inequality of Paley and convergence a.e. of Walsh-Fourier series. Ark. Mat. 7, 551–570 (1969)

    Google Scholar 

  35. S.B. Stechkin, On the convergence and divergence of trigonometric series. Uspekhi Mat. Nauk (in Russian) 6(2), 148–149 (1951)

    Google Scholar 

  36. E.M. Stein, On limits of sequences of operators. Ann. Math. 74(2), 140–170 (1961)

    Article  MathSciNet  Google Scholar 

  37. L.V. Taukov On the divergence of Fourier series with respect to a rearranged trigonometric system (in Russian). Uspekhi Mat. Nauk 18(5), 191–198 (1963)

    Google Scholar 

  38. G. Tephnadze, Martingale Hardy spaces and summability of the one dimensional Vilenkin-Fourier series, Ph.D. thesis, Department of Engineering Sciences and Mathematics, Luleå University of Technology, ISSN 1402–1544, 2015

    Google Scholar 

  39. A. Zygmund, Trigonometric Series, vols. I, II (Cambridge University Press, 1959)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University, Tbilisi, Georgia

    Nika Areshidze

  2. UiT The Arctic University of Norway, Narvik, Norway

    Lars-Erik Persson

  3. Department of Mathematics and Computer Science, Karlstad University, Karlstad, Sweden

    Lars-Erik Persson

  4. School of Science and Technology, The University of Georgia, Tbilisi, Georgia

    George Tephnadze

Authors
  1. Nika Areshidze
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lars-Erik Persson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. George Tephnadze
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Mathematics, University of Georgia, Tbilisi, Georgia

    Roland Duduchava

  2. Department of Mathematics, King's College London, London, UK

    Eugene Shargorodsky

  3. Institute of Mathematics, University of Georgia, Tbilisi, Georgia

    George Tephnadze

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Areshidze, N., Persson, LE., Tephnadze, G. (2024). On Divergence of Fejér Means with Respect to Walsh System on Sets of Measure Zero. In: Duduchava, R., Shargorodsky, E., Tephnadze, G. (eds) Tbilisi Analysis and PDE Seminar. TAPDES 2023. Trends in Mathematics(), vol 7. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-62894-8_3

Download citation

Publish with us

Policies and ethics

Search

Navigation