Abstract
We give a brief overview of the results on the behavior of the Lebesgue constants for various partial sums of multiple Fourier series. In addition, we establish a new property of the Lebesgue constants concerning its partly increasing behavior.
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References
L. Aizenberg, N. Tarkhanov, Stable expansions in homogeneous polynomials, Complex analysis and dynamical systems III. Contemp. Math. 455, 1–22 (2008). Israel Math. Conf. Proc., Amer. Math. Soc., Providence, RI
K.I. Babenko, Approximation by trigonometric polynomials in a certain class of periodic functions of several variables. Dokl. Akad. Nauk SSSR 132, 982–985 (1960) (Russian). English transl. in Soviet Math. Dokl. 1, 672–675 (1960)
K.I. Babenko, On the mean convergence of multiple Fourier series and the asymptotics of the Dirichlet kernel of spherical means. Inst. Prikl. Mat. Akad. Nauk SSSR, Moscow, Preprint No. 52 (1971) (Russian)
K.I. Babenko, On the asymptotics of the Dirichlet kernel of spherical means of multiple Fourier series. Dokl. Akad. Nauk SSSR 243, 1097–1100 (1978) (Russian). English transl. in Soviet Math. Dokl. 19, 1457–1461 (1978)
S.P. Baiborodov, Lebesgue constants of polyhedra. Mat. Zametki 32, 817–822 (1982) (Russian). English transl. in Math. Notes 32, 895–898 (1982)
E. Belinsky, Behavior of the Lebesgue constants of certain methods of summation of multiple Fourier series, in Metric Questions of the Theory of Functions and Mappings (Naukova Dumka, Kiev, 1977), pp. 19–39 (Russian)
E. Belinsky, Some properties of hyperbolic partial sums of Fourier series and Fourier integrals. Dokl. Acad. Sci. Ukraine Ser. A 10, 869–870 (1978) (Russian)
E. Belinsky, Some properties of hyperbolic partial sums, in Theory of Functions and Mappings (Naukova Dumka, Kiev, 1979), pp. 28–36 (Russian)
E. Belinsky, On the growth of Lebesgue constants of partial sums generated by certain unbounded sets, in Theory of Mappings and Approximation of Functions (Naukova Dumka, Kiev, 1983), pp. 18–20 (Russian)
E. Belinsky, E. Liflyand, Behavior of the Lebesgue constants of hyperbolic partial sums. Mat. Zametki 43, 192–196 (1988) (Russian). English transl. in Math. Notes 43 (1988), 107–109
L. Brandolini, Fourier transform of characteristic functions and Lebesgue constants for multiple Fourier series. Colloquium Math. LXV, 51–59 (1993)
M. Carenini, P.M. Soardi, Sharp estimates for Lebesgue constants. Proc. Am. Math. Soc. 89, 449–452 (1983)
L. Colzani, P.M. Soardi, L p norms of certain kernels on the N-dimensional torus. Trans. Am. Math. Soc. 266, 617–627 (1981)
I.K. Daugavet, On the Lebesgue constants for double Fourier series. Meth. Comput. Leningrad Univ. 6, 8–13 (1970) (Russian)
P. Dencker, W. Erb, Y. Kolomoitsev, T. Lomako, Lebesgue constants for polyhedral sets and polynomial interpolation on Lissajous-Chebyshev nodes. J. Complexity 43, 1–27 (2017)
M. Dyachenko, Some problems in the theory of multiple trigonometric series. Uspekhi Mat. Nauk 47(5), 97–162 (1992) (Russian). English transl. in Russian Math. Surveys 47(5), 103–171 (1992)
C. Fefferman, On the divergence of multiple Fourier series. Bull. Am. Math. Soc. 77, 191–195 (1971)
L. Fejér, Lebesguessche Konstanten und divergente Fourierreihen. J. Reine Angew. Math. 138, 22–53 (1910)
S. Giulini, G. Travaglini, Sharp estimates for Lebesgue constants on compact lie groups. J. Funct. Anal. 68, 106–116 (1986)
E. Gutierrez Gonzales, A lower bound for two-dimensional Lebesgue constants. Vestnik S.-Peterburg. Univ. Mat. 26, 119–121 (1993) (Russian). English transl. in Vestnik St. Peterburg Univ. Math. 26, 69–71 (1993)
V.A. Ilyin, Problems of localization and convergence for Fourier series in fundamental systems of the Laplace operator. Uspekhi Mat. Nauk 23, 61–120 (1968) (Russian). English transl. in Russian Math. Surveys 23 (1968), 59–116
V.A. Ilyin, S.A. Alimov, Conditions for the convergence of expansions corresponding to self-adjoint extensions of elliptic operators. I. Diff. Urav. 7, 670–710 (1971) (Russian). English transl. in Differential Equations 7, 516–543 (1971)
A. Iosevich, E. Liflyand, Decay of the Fourier Transform: Analytic and Geometric Aspects (Birkhauser, Basel, 2014)
Y. Kolomoitsev, T. Lomako, On the growth of Lebesgue constants for convex polyhedra. Trans. Am. Math. Soc. (to appear). https://doi.org/10.1090/tran/7225; see also https://arxiv.org/abs/1801.00608
O. Kuznetsova, The asymptotic behavior of the Lebesgue constants for a sequence of triangular partial sums of double Fourier series. Sib. Mat. Zh. XVIII, 629–636 (1977) (Russian). English transl. in Siberian Math. J. 18, 449–454 (1977)
E. Liflyand, Exact order of the Lebesgue constants of hyperbolic partial sums of multiple Fourier series. Mat. Zametki 39, 674–683 (1986) (Russian). English transl. in Math. Notes 39, 369–374 (1986)
E. Liflyand, Sharp estimates of the Lebesgue constants of partial sums of multiple Fourier series. Trudy Mat. Inst. Im. V. A. Steklova 180, 151–152 (1987) (Russian). English transl. in Proc. Steklov Math. Inst. 180, 176–177 (1989)
E. Liflyand, Lebesgue constants of Cesà ro means for spherical harmonic expansions. Acta Sci. Math. (Szeged) 64, 215–222 (1998)
E. Liflyand, Lebesgue constants of multiple Fourier series. Online J. Anal. Combin. 1, 112 (2006)
E.R. Liflyand, A.G. Ramm, A.I. Zaslavsky, Estimates from below for Lebesgue constants. J. Fourier Anal. Appl. 2, 287–301 (1996)
J. Marshall Ash, L. De Carli, Growth of L p Lebesgue constants for convex polyhedra and other regions. Trans. Am. Math. Soc. 361, 4215–4232 (2009)
F. Nazarov, A. Podkorytov, On the behavior of the Lebesgue constants for two-dimensional Fourier sums over polygons. Algebra i Analiz 7, 214–238 (1995) (Russian). English transl. in St.-Petersburg Math. J. 7, 663–680 (1995)
A.N. Podkorytov, On the Lebesgue constants for double Fourier series. Vestnik Leningrad. Univ. Matem. 7, 79–84 (1977) (Russian). English transl. in Vestnik Leningrad Univ. Math. 10, 187–192 (1982)
A.N. Podkorytov, The order of growth of the Lebesgue constants of Fourier sums over polyhedra. Vestn. Leningr. Univ. Matem. 7, 110–111 (1982) (Russian)
A.N. Podkorytov, Intermediate rates of growth of Lebesgue constants in the two-dimensional case, Numerical methods and questions on the organization of calculations, part 7 notes of scientific seminars of the V.A. Steklov Math. Inst. Leningrad. Branch, Acad. Sci. USSR, Nauka, Leningrad 139, 148–155 (1984) (Russian). English transl. in J. Soviet Math. 32, 276–282 (1987)
A.N. Podkorytov, Asymptotic behavior of the Dirichlet kernel of Fourier sums with respect to a polygon. Zap. Nauch. Sem. LOMI 149, 142–149 (1986) (Russian). English transl. in J. Soviet Math. 42, 1640–1646 (1988)
W. Rudin, Fourier analysis on groups, in Interscience Tracts in Pure and Applied Mathematics, No. 12 (Interscience Publishers (a division of John Wiley and Sons), New York, 1962)
H.S. Shapiro, Lebesgue constants for spherical partial sums. J. Appr. Theory 13, 40–44 (1975)
M.A. Skopina, Lebesgue constants of multiple de la Vallée Poussin sums over polyhedra. Zap. Nauch. Sem. LOMI 125, 154–165 (1983) (Russian). English transl. in J. Soviet Math. 26, 2404–2413 (1984)
G. Szegö, Über die Lebesgueschen Konstanten bei den Fourierschen Reihen. Math. Zeitschrift 9, 163–166 (1921)
R.M. Trigub, E.S. Belinsky, Fourier Analysis and Approximation of Functions (Springer, Netherlands, 2004)
V.A. Yudin, Behavior of Lebesgue constants. Mat. Zametki 17, 401–405 (1975) (Russian). English transl. in Math. Notes 17, 369–374 (1975)
V.A. Yudin, A lower bound for Lebesgue constants. Mat. Zametki 25, 119–122 (1979) (Russian). English transl. in Math. Notes 25, 63–65 (1979)
A.A. Yudin, V.A. Yudin, Discrete imbedding theorems and Lebesgue constants. Mat. Zametki 22, 381–394 (1977) (Russian). English transl. in Math. Notes 22, 702–711 (1977)
A.A. Yudin, V.A. Yudin, Polygonal Dirichlet Kernels and growth of Lebesgue constants. Mat. Zametki 37, 220–236 (1985) (Russian). English transl. in Math. Notes 37, 124–135 (1985)
A. Zygmund, Trigonometric Series: Vols. I, II, Second edition, reprinted with corrections and some additions (Cambridge University Press, London, 1968)
Acknowledgements
The authors acknowledge the support of the Gelbart Institute at the Mathematics Department in Bar-Ilan University.
The authors are indebted to the referee for thorough reading and valuable remarks and suggestions.
Last but not least, the authors thank A. Podkorytov for bringing their attention to Szegö’s result and for interesting discussions.
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Ganzburg, M.I., Liflyand, E. (2019). The Lebesgue Constants of Fourier Partial Sums. In: Abell, M., Iacob, E., Stokolos, A., Taylor, S., Tikhonov, S., Zhu, J. (eds) Topics in Classical and Modern Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-12277-5_10
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