Abstract
We introduce a new concept of Lebesgue points, the so-called w-Lebesgue points, where w > 0. As a generalization of the classical Lebesgue’s theorem, we prove that the Cesàro means ρanf of the Fourier series of a multidimensional function f ∈ L1(Td) converge to f at each w-Lebesgue point (0 < w < α) as n → ∞.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H.G. Feichtinger and F. Weisz, Wiener amalgams and pointwise summability of Fourier transforms and Fourier series, Math. Proc. Cambridge Philos. Soc., 140 (2006), 509–536.
L. Fejér, Untersuchungen über Fouriersche Reihen, Math. Ann., 58 (1904), 51–69.
O.D. Gabisoniya, Points of summability of double Fourier series by certain linear methods, Izv. Vyssh. Uchebn. Zaved., Mat., 5(120) (1972), 29–37 (in Russian).
G. Gát, Pointwise convergence of cone-like restricted two-dimensional (C, 1) means of trigonometric Fourier series, J. Approx. Theory., 149 (2007), 74–102.
G. Gát, Almost everywhere convergence of sequences of Cesàro and Riesz means of integrable functions with respect to the multidimensional Walsh system, Acta Math. Sin., Engl. Ser., 30 (2014), 311–322.
G. Gát, U. Goginava and K. Nagy, On the Marcinkiewicz–Fejér means of double Fourier series with respect to Walsh-Kaczmarz system, Studia Sci. Math. Hungar., 46 (2009), 399–421.
U. Goginava, Marcinkiewicz–Fejér means of d-dimensional Walsh–Fourier series, J. Math. Anal. Appl., 307 (2005), 206–218.
U. Goginava, Almost everywhere convergence of (C, α)-means of cubical partial sums of d-dimensional Walsh-Fourier series, J. Approx. Theory, 141 (2006), 8–28.
U. Goginava, The maximal operator of the Marcinkiewicz–Fejér means of d- dimensional Walsh–Fourier series, East J. Approx., 12 (2006), 295–302.
H. Lebesgue, Recherches sur la convergence des séries de Fourier, Math. Ann., 61 (1905), 251–280.
L. Leindler, Strong approximation by Fourier series, Akadémiai Kiadó, Budapest, 1985.
J. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series, Fund. Math., 32 (1939), 122–132.
K. Nagy and G. Tephnadze, The Walsh-Kaczmarz-Marcinkiewicz means and Hardy spaces, Acta Math. Hungar., 149 (2016), 346–374.
L. E. Persson, G. Tephnadze and P. Wall, Maximal operators of Vilenkin- Nörlund means, J. Fourier Anal. Appl., 21 (2015), 76–94.
M. Riesz, Sur la sommation des séries de Fourier, Acta Sci. Math. (Szeged), 1 (1923), 104–113.
P. Simon, Cesàro summability with respect to two-parameter Walsh systems, Monatsh. Math., 131 (2000), 321–334.
P. Simon, (C, α) summability of Walsh-Kaczmarz-Fourier series, J. Approx. Theory, 127 (2004), 39–60.
M. A. Skopina, The generalized Lebesgue sets of functions of two variables, Approximation theory. Proceedings of a conference organized by the János Bolyai Mathematical Society, held in Kecskemét, Hungary, August 6 to 11, 1990, North-Holland Publishing Company, Amsterdam; János Bolyai Mathematical Society, Budapest, 1991, 615–625.
M. A. Skopina, The order of growth of quadratic partial sums of a double Fourier series, Math. Notes, 51 (1992), 1.
F. Weisz, Summability of Multi-dimensional Fourier Series and Hardy Spaces, Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht–Boston–London, 2002.
F. Weisz, Summability of multi-dimensional trigonometric Fourier series, Surv. Approx. Theory, 7 (2012), 1–179.
F. Weisz, Lebesgue points of two-dimensional Fourier transforms and strong summability, J. Fourier Anal. Appl., 21 (2015), 885–914.
F. Weisz, Lebesgue points and restricted convergence of Fourier transforms and Fourier series, Anal. Appl. (Singap.), 15 (2017), 107–121.
A. Zygmund, Trigonometric Series, 3rd edition, Cambridge University Press, London, 2002.
Acknowledgment
I would like to thank the referee for reading the paper carefully and for his/her useful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
To the memory of Professor László Leindler
Communicated by L. Molnár
This research was supported by the Hungarian Scientific Research Funds (OTKA) No KH130426.
Rights and permissions
About this article
Cite this article
Weisz, F. Lebesgue points and Cesàro summability of higher dimensional Fourier series over a cone. ActaSci.Math. 87, 505–515 (2021). https://doi.org/10.14232/actasm-021-614-3
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.14232/actasm-021-614-3