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Korovkin-type approximation of set-valued and vector-valued functions
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 |
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 $.
References:
[1] |
J. Arias de Reyna,
Pointwise convergence of fourier series, J. London Math. Soc., 65 (2002), 139-153.
doi: 10.1112/S0024610701002824. |
[2] |
N. K. Bary, A Treatise on Trigonometric Series, Vols. I, II. Authorized translation by Margaret F. Mullins. A Pergamon Press Book The Macmillan Company, New York 1964. |
[3] |
E. S. Belinsky, Summability of multiple Fourier series at Lebesgue points, Teor. Funkci$\mathop l\limits^ \vee $ Funkcional. Anal. i Priložen, 169 (1975), 3–12, (Russian). |
[4] |
H. Berens, Z. 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 $S_0(\mathbb R^d)$ 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. 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.
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. Jessen, J. 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. Persson, G. 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. Berens, Z. 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 $S_0(\mathbb R^d)$ 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. 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.
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. Jessen, J. 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. Persson, G. 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.
![]() ![]() |

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