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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
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)
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)
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)
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)
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)
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)
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)
L. Carleson, On convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157 (1966)
C. Demeter, A guide to Carleson’s theorem. Rocky Mountain J. Math. 45(1), 169–212 (2015)
Ch. Fefferman, Pointwise convergence of Fourier series. Ann. Math. 98(3), 551–571 (1973)
D.H. Fremlin, Measure Theory, vol. 2, Broad Foundations. Corrected second printing of the 2001 original (Torres Fremlin, Colchester, 2003)
U. Goginava, On divergence of Walsh-Fejer means of bounded functions on sets of measure zero. Acta Math. Hungar. 121(3), 359–369 (2008)
L. Grafakos, Classical Fourier Analysis, 3rd edn, vol. 249. Graduate Texts in Mathematics (Springer, New York, 2014)
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
J.-P. Kahane, Y. Katznelson, Sur les ensembles de divergence des séries trigonométriques. Stud. Math. 26, 305–306 (1966)
G.A. Karagulyan, Divergence of general operators on sets of measure zero. Colloquium Math. 121(1), 113–119 (2010)
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)
Sh.V. Kheladze, On the everywhere divergence of Fourier-Walsh series. Soobshch. Akad. Nauk. Gruzin. 77, 305–307 (1975)
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)
A.N. Kolmogorov, Une série de Fourier-Lebesgue divergente presque partout. Polska Akad. Nauk. Fund. Math. 4, 324–328 (1923)
M.T. Lacey, Carleson’s theorem: proof, complements, variations. Publ. Mat. 48(2), 251–307 (2004)
M. Lacey, C. Thiele, A proof of boundedness of the Carleson operator. Math. Res. Lett. 7(4), 361–370 (2000)
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)
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)
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)
L.-E. Persson, G. Tephnadze, P. Wall, On the maximal operators of Vilenkin-Nörlund means. J. Fourier Anal. Appl. 21(1), 76–94 (2015)
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)
L.-E. Persson, G. Tephnadze, F. Weisz, Martingale Hardy Spaces and Summability of One-dimensional Vilenkin-Fourier Series (Birkhäuser/Springer, 2022)
F. Schipp, Über die divergenz der Walsh-Fourierreihen. Ann. Univ. Sci. Budapest Eötvös Sect. Math. 12, 49–62 (1969)
F. Schipp, Certain rearrangements of series in the Walsh series. Mat. Zametki 18, 193–201 (1975)
F. Schipp, Pointwise convergence of expansions with respect to certain product systems. Anal. Math. 2(1), 65–76 (1976)
F. Schipp, W.R. Wade, P. Simon, J. Pál, Walsh series. An Introduction to Dyadic Harmonic Analysis (Adam Hilger, Bristol, 1990)
P. Sjölin, An inequality of Paley and convergence a.e. of Walsh-Fourier series. Ark. Mat. 7, 551–570 (1969)
S.B. Stechkin, On the convergence and divergence of trigonometric series. Uspekhi Mat. Nauk (in Russian) 6(2), 148–149 (1951)
E.M. Stein, On limits of sequences of operators. Ann. Math. 74(2), 140–170 (1961)
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)
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
A. Zygmund, Trigonometric Series, vols. I, II (Cambridge University Press, 1959)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
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
DOI: https://doi.org/10.1007/978-3-031-62894-8_3
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-031-62893-1
Online ISBN: 978-3-031-62894-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)