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

Year 2021, Volume: 4 Issue: 2, 179 - 185, 01.06.2021

Ferenc Weisz

https://doi.org/10.33205/cma.859583

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

• 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.
• 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.
• L. Fejér: Untersuchungen über Fouriersche Reihen, Math. Ann., 58 (1904), 51–69.
• 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., 30 (2) (2014), 311–322.
• 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.
• 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.
• B. Jessen, J. Marcinkiewicz and A. Zygmund: Note on the differentiability of multiple integrals, Fundam. Math., 25 (1935), 217–234.
• H. Lebesgue: Recherches sur la convergence des séries de Fourier, Math. Ann., 61 (1905), 251–280.
• J. Marcinkiewicz, A. Zygmund: On the summability of double Fourier series, Fund. Math., 32 (1939), 122–132.
• K. Nagy, 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 (1) (2015), 76–94.
• M. Riesz: Sur la sommation des séries de Fourier, Acta Sci. Math. (Szeged), 1 (1923), 104–113.
• S. Saks: Remark on the differentiability of the Lebesgue indefinite integral, Fundam. Math., 22 (1934) 257–261.
• 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.
• 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, .
• A. Zygmund: Trigonometric Series. Cambridge Press, London, 3rd edition, (2002).