August  2022, 5(3): 241-257. doi: 10.3934/mfc.2021033

Cesàro summability and Lebesgue points of higher dimensional Fourier series

Department of Numerical Analysis, Eötvös L. University, Pázmány P. sétány 1/C, H-1117 Budapest, Hungary

Corresponding author: Ferenc Weisz

Received  June 2021 Revised  October 2021 Published  August 2022 Early access  November 2021

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

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 $.

Citation: Ferenc Weisz. Cesàro summability and Lebesgue points of higher dimensional Fourier series. Mathematical Foundations of Computing, 2022, 5 (3) : 241-257. doi: 10.3934/mfc.2021033
References:
[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. Chang, Lectures 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 $S_0(\mathbb mathbb{R}^{d})$ 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. Schlag, Classical 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. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, N. J., 1993. 
[48] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N. J., 1971. 
[49] A. Torchinsky, Real-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. Zygmund, Trigonometric Series, 2$^{nd}$ edition, Cambridge Press, London, 1968. 

show all references

References:
[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. Chang, Lectures 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 $S_0(\mathbb mathbb{R}^{d})$ 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. Schlag, Classical 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. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, N. J., 1993. 
[48] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N. J., 1971. 
[49] A. Torchinsky, Real-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. Zygmund, Trigonometric Series, 2$^{nd}$ edition, Cambridge Press, London, 1968. 
Figure 1.  Regions of the $ \ell_q $-partial sums for $ d = 2 $
Figure 2.  The cone for $ d = 2 $
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