Advanced Search
Article Contents
Article Contents

Cesàro summability and Lebesgue points of higher dimensional Fourier series

  • Corresponding author: Ferenc Weisz

    Corresponding author: Ferenc Weisz

This research was supported by the Hungarian Scientific Research Funds (OTKA) No KH130426

Abstract Full Text(HTML) Figure(2) Related Papers Cited by
  • We give four generalizations of the classical Lebesgue's theorem to multi-dimensional functions and Fourier series. We introduce four new concepts of Lebesgue points, the corresponding Hardy-Littlewood type maximal functions and show that almost every point is a Lebesgue point. For four different types of summability and convergences investigated in the literature, we prove that the Cesàro means $ \sigma_n^{\alpha}f $ of the Fourier series of a multi-dimensional function converge to $ f $ at each Lebesgue point as $ n\to \infty $.

    Mathematics Subject Classification: Primary: 42B08; Secondary 42A24, 42B25.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Regions of the $ \ell_q $-partial sums for $ d = 2 $

    Figure 2.  The cone for $ d = 2 $

  • [1] J. Arias de Reyna, Pointwise convergence of fourier series, J. London Math. Soc., 65 (2002), 139-153.  doi: 10.1112/S0024610701002824.
    [2] N. K. Bary, A Treatise on Trigonometric Series, Vols. I, II. Authorized translation by Margaret F. Mullins. A Pergamon Press Book The Macmillan Company, New York 1964.
    [3] E. S. Belinsky, Summability of multiple Fourier series at Lebesgue points, Teor. Funkci$\mathop l\limits^ \vee $ Funkcional. Anal. i Priložen, 169 (1975), 3–12, (Russian).
    [4] H. BerensZ. Li and Y. Xu, On $l_1$ Riesz summability of the inverse Fourier integral, Indag. Math. (N.S.), 12 (2001), 41-53.  doi: 10.1016/S0019-3577(01)80004-5.
    [5] H. Berens and Y. Xu, Fejér means for multivariate Fourier series, Math. Z., 221 (1996), 449-465.  doi: 10.1007/PL00004254.
    [6] H. Berens and Y. Xu, $l$-1 summability of multiple Fourier integrals and positivity, Math. Proc. Cambridge Philos. Soc., 122 (1997), 149-172.  doi: 10.1017/S0305004196001521.
    [7] L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math., 116 (1966), 135-157.  doi: 10.1007/BF02392815.
    [8] S. Y. A. Chang and R. Fefferman, Some recent developments in Fourier analysis and $H^p$-theory on product domains, Bull. Amer. Math. Soc., 12 (1985), 1-43.  doi: 10.1090/S0273-0979-1985-15291-7.
    [9] K. M. Davis and  Y. C. ChangLectures on Bochner-Riesz Means, vol. 114 of London Mathematical Society Lecture Note Series, Cambridge University Press, 1987.  doi: 10.1017/CBO9781107325654.
    [10] C. Demeter, A guide to Carleson's theorem, Rocky Mt. J. Math., 45 (2015), 169-212.  doi: 10.1216/RMJ-2015-45-1-169.
    [11] P. du Bois-Reymond, Convergenz und Divergenz der Fourier'schen Darstellungsformeln, Math. Ann., 10 (1876), 431-445.  doi: 10.1007/BF01442324.
    [12] C. Fefferman, On the convergence of multiple Fourier series, Bull. Amer. Math. Soc., 77 (1971), 744-745.  doi: 10.1090/S0002-9904-1971-12793-3.
    [13] C. Fefferman, On the divergence of multiple Fourier series, Bull. Amer. Math. Soc., 77 (1971), 191-195.  doi: 10.1090/S0002-9904-1971-12675-7.
    [14] C. Fefferman, The multiplier problem for the ball, Ann. of Math., 94 (1971), 330-336.  doi: 10.2307/1970864.
    [15] H. G. Feichtinger and F. Weisz, The Segal algebra $S_0(\mathbb R^d)$ and norm summability of Fourier series and Fourier transforms, Monatsh. Math., 148 (2006), 333-349.  doi: 10.1007/s00605-005-0358-4.
    [16] 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.  doi: 10.1017/S0305004106009273.
    [17] L. Fejér, Untersuchungen über fouriersche reihen, Math. Ann., 58 (1903), 51-69.  doi: 10.1007/BF01447779.
    [18] L. Fejér, Beispiele stetiger Funktionen mit divergenter Fourier-reihe, J. Reine Angew. Math., 137 (1910), 1-5.  doi: 10.1515/crll.1910.137.1.
    [19] O. D. Gabisoniya, Points of summability of double Fourier series by certain linear methods, Izv. Vyssh. Uchebn. Zaved., Mat., 5 (1972), 29–37, (Russian).
    [20] G. Gát, Pointwise convergence of cone-like restricted two-dimensional $(C, 1)$ means of trigonometric Fourier series, J. Approx. Theory., 149 (2007), 74-102.  doi: 10.1016/j.jat.2006.08.006.
    [21] 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.  doi: 10.1007/s10114-013-1766-3.
    [22] G. GátU. 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.  doi: 10.1556/sscmath.2009.1099.
    [23] U. Goginava, Marcinkiewicz-Fejér means of $d$-dimensional Walsh-Fourier series, J. Math. Anal. Appl., 307 (2005), 206-218.  doi: 10.1016/j.jmaa.2004.11.001.
    [24] U. Goginava, Almost everywhere convergence of $(C, \alpha)$-means of cubical partial sums of d-dimensional Walsh-Fourier series, J. Approx. Theory, 141 (2006), 8-28.  doi: 10.1016/j.jat.2006.01.001.
    [25] U. Goginava, The maximal operator of the Marcinkiewicz-Fejér means of $d$-dimensional Walsh-Fourier series, East J. Approx., 12 (2006), 295-302. 
    [26] L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, New Jersey, 2004.
    [27] L. Grafakos, Classical Fourier Analysis, 3$^{rd}$ edition, Graduate Texts in Mathematics, 249. Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.
    [28] L. Grafakos, Modern Fourier Analysis, 3$^{rd}$ edition, Graduate Texts in Mathematics, 250. Springer, New York, 2014. doi: 10.1007/978-1-4939-1230-8.
    [29] R. A. Hunt, On the convergence of Fourier series, In Orthogonal Expansions and Their Continuous Analogues, Proc. Conf. Edwardsville, Ill., 1967, Illinois Univ. Press Carbondale, (1967), 235–255.
    [30] B. JessenJ. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals, Fundam. Math., 25 (1935), 217-234. 
    [31] A. N. Kolmogorov, Un serie de Fourier-Lebesgue divergente presque partout, Fundamenta Math., 4 (1923), 324-328. 
    [32] A. N. Kolmogorov, Un serie de Fourier-Lebesgue divergente partout, C. R. Acad. Sci. Pariss, 183 (1926), 1327-1328. 
    [33] M. T. Lacey, Carleson's theorem: Proof, complements, variations, Publ. Mat., Barc., 48 (2004), 251-307. 
    [34] H. Lebesgue, Recherches sur la convergence des séries de Fourier, Math. Ann., 61 (1905), 251-280.  doi: 10.1007/BF01457565.
    [35] S. Lu and D. Yan, Bochner-Riesz Means on Euclidean Spaces, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. doi: 10.1142/8745.
    [36] J. Marcinkiewicz, Sur une méthode remarquable de sommation des séries doubles de Fourier, Ann. Scuola Norm. Sup. Pisa, 8 (1939), 149-160. 
    [37] J. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series, Fund. Math., 32 (1939), 122-132. 
    [38] C. Muscalu and  W. SchlagClassical and Multilinear Harmonic Analysis, Cambridge University Press, Cambridge, 2013. 
    [39] K. Nagy and G. Tephnadze, The Walsh-Kaczmarz-Marcinkiewicz means and Hardy spaces, Acta Math. Hungar., 149 (2016), 346-374.  doi: 10.1007/s10474-016-0617-y.
    [40] L. E. PerssonG. Tephnadze and P. Wall, Maximal operators of Vilenkin-Nörlund means, J. Fourier Anal. Appl., 21 (2015), 76-94.  doi: 10.1007/s00041-014-9345-2.
    [41] M. Riesz, Sur la sommation des séries de Fourier, Acta Sci. Math. (Szeged), 1 (1923), 104-113. 
    [42] S. Saks, Remark on the differentiability of the Lebesgue indefinite integral, Fundam. Math., 22 (1934), 257-261. 
    [43] P. Simon, Cesàro summability with respect to two-parameter Walsh systems, Monatsh. Math., 131 (2000), 321-334.  doi: 10.1007/s006050070004.
    [44] P. Simon, $(C, \alpha)$ summability of Walsh-Kaczmarz-Fourier series, J. Approx. Theory, 127 (2004), 39-60.  doi: 10.1016/j.jat.2004.02.003.
    [45] M. A. Skopina, The generalized Lebesgue sets of functions of two variables, Approximation theory, Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam, 58 (1991), 615-625. 
    [46] M. A. Skopina, The order of growth of quadratic partial sums of a double Fourier series, Math. Notes, 51 (1992), 576-582.  doi: 10.1007/BF01263302.
    [47] E. M. SteinHarmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, N. J., 1993. 
    [48] E. M. Stein and  G. WeissIntroduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N. J., 1971. 
    [49] A. TorchinskyReal-variable Methods in Harmonic Analysis, Academic Press, Inc., Orlando, FL, 1986. 
    [50] F. Weisz, $(C, \alpha)$ means of $d$-dimensional trigonometric-Fourier series, Publ. Math. Debrecen, 52 (1998), 705-720. 
    [51] F. Weisz, Summability of multi-dimensional trigonometric Fourier series, Surv. Approx. Theory, 7 (2012), 1-179. 
    [52] F. Weisz, Lebesgue points of two-dimensional Fourier transforms and strong summability, J. Fourier Anal. Appl., 21 (2015), 885-914.  doi: 10.1007/s00041-015-9393-2.
    [53] F. Weisz, Convergence and Summability of Fourier Transforms and Hardy Spaces, Applied and Numerical Harmonic Analysis, Springer, Birkhäuser, Basel, 2017.
    [54] F. Weisz, Marcinkiewicz summability of Fourier series, Lebesgue points and strong summability, Acta Math. Hungar., 153 (2017), 356-381.  doi: 10.1007/s10474-017-0737-z.
    [55] F. Weisz, Lebesgue points and Cesàro summability of higher dimensional Fourier series over a cone, Acta Sci. Math. (Szeged), 87 (2021), 505-515. 
    [56] F. Weisz, Lebesgue points of $\ell_1$-Cesàro summability of $d$-dimensional Fourier series, Adv. Oper. Theory., 6 (2021), 48.  doi: 10.1007/s43036-021-00144-3.
    [57] F. Weisz, Unrestricted Cesàro summability of $d$-dimensional Fourier series and Lebesgue points, Constr. Math. Anal., 4 (2021), 179-185. 
    [58] Y. Xu, Christoffel functions and Fourier series for multivariate orthogonal polynomials, J. Approx. Theory, 82 (1995), 205-239.  doi: 10.1006/jath.1995.1075.
    [59] L. Zhizhiashvili, Trigonometric Fourier Series and their Conjugates, Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-94-009-0283-1.
    [60] A. ZygmundTrigonometric Series, 2$^{nd}$ edition, Cambridge Press, London, 1968. 
  • 加载中



Article Metrics

HTML views(628) PDF downloads(391) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint