Unrestricted Cesàro summability of $d$-dimensional Fourier series and Lebesgue points

Abstract

We generalize the classical Lebesgue's theorem to multi-dimensional functions. We prove that the Cesàro means of the Fourier series of the multi-dimensional function $f\in L_1(\log L)^{d-1}(\mathbb{T}^d)\supset L_p(\mathbb{T}^d) (1<p<\infty)$ converge to $f$ at each strong Lebesgue point.

Keywords

• Cesàro summability
• strong Hardy-Littlewood maximal function
• strong Lebesgue points

References

1. S. Y. A. Chang, R. Fefferman: Some recent developments in Fourier analysis and Hp-theory on product domains, Bull. Amer. Math. Soc., 12 (1985), 1–43.
2. H. G. Feichtinger, F. Weisz: Wiener amalgams and pointwise summability of Fourier transforms and Fourier series, Math. Proc. Cambridge Philos. Soc., 140 (2006), 509–536.
3. L. Fejér: Untersuchungen über Fouriersche Reihen, Math. Ann., 58 (1904), 51–69.
4. G. Gát: Pointwise convergence of cone-like restricted two-dimensional (C, 1) means of trigonometric Fourier series, J. Approx. Theory., 149 (2007), 74–102.
5. 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., 30 (2) (2014), 311–322.
6. G. Gát, U. Goginava and K. Nagy: On the Marcinkiewicz-Fejér means of double Fourier series with respect to WalshKaczmarz system, Studia Sci. Math. Hungar., 46 (2009), 399–421.
7. U. Goginava: Marcinkiewicz-Fejér means of d-dimensional Walsh-Fourier series, J. Math. Anal. Appl., 307 (2005), 206–218.
8. 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.

9. U. Goginava: The maximal operator of the Marcinkiewicz-Fejér means of d-dimensional Walsh-Fourier series, East J. Approx., 12 (2006), 295–302.
10. B. Jessen, J. Marcinkiewicz and A. Zygmund: Note on the differentiability of multiple integrals, Fundam. Math., 25 (1935), 217–234.
11. H. Lebesgue: Recherches sur la convergence des séries de Fourier, Math. Ann., 61 (1905), 251–280.
12. J. Marcinkiewicz, A. Zygmund: On the summability of double Fourier series, Fund. Math., 32 (1939), 122–132.
13. K. Nagy, G. Tephnadze: The Walsh-Kaczmarz-Marcinkiewicz means and Hardy spaces, Acta Math. Hungar., 149 (2016), 346–374.
14. L. E. Persson, G. Tephnadze and P. Wall: Maximal operators of Vilenkin-Nörlund means, J. Fourier Anal. Appl., 21 (1) (2015), 76–94.
15. M. Riesz: Sur la sommation des séries de Fourier, Acta Sci. Math. (Szeged), 1 (1923), 104–113.
16. S. Saks: Remark on the differentiability of the Lebesgue indefinite integral, Fundam. Math., 22 (1934) 257–261.
17. P. Simon: Cesàro summability with respect to two-parameter Walsh systems, Monatsh. Math., 131 (2000), 321–334.
18. P. Simon: (C, α) summability of Walsh-Kaczmarz-Fourier series, J. Approx. Theory, 127 (2004), 39–60.
19. F. Weisz: Summability of Multi-dimensional Fourier Series and Hardy Spaces, Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, (2002).
20. F. Weisz: Summability of multi-dimensional trigonometric Fourier series, Surv. Approx. Theory, 7 (2012), 1–179, .
21. A. Zygmund: Trigonometric Series. Cambridge Press, London, 3rd edition, (2002).