We study polyhedral Dirichlet kernels on the n-dimensional torus and we write a fairly simple formula which extends the one-dimensional identity . We prove sharp results for the Lebesgue constants and for the pointwise boundedness of polyhedral Dirichlet kernels; we apply our results and methods to approximation theory, to more general summability methods and to Fourier series on compact Lie groups, where we write an asymptotic formula for the Dirichlet kernels.
@article{bwmeta1.element.bwnjournal-article-cmv65i1p103bwm,
author = {Giancarlo Travaglini},
title = {Polyhedral summability of multiple Fourier series (and explicit formulas for Dirichlet kernels on $^n$ and on compact Lie groups)},
journal = {Colloquium Mathematicae},
volume = {66},
year = {1993},
pages = {103-116},
zbl = {0818.42004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv65i1p103bwm}
}
Travaglini, Giancarlo. Polyhedral summability of multiple Fourier series (and explicit formulas for Dirichlet kernels on $^n$ and on compact Lie groups). Colloquium Mathematicae, Tome 66 (1993) pp. 103-116. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv65i1p103bwm/
[000] [1] S. A. Alimov, V. A. Il'in and E. M. Nikishin, Convergence problems for multiple trigonometric series and spectral decomposition, Russian Math. Surveys 31 (1976), 29-86. | Zbl 0367.42008
[001] [2] L. Brandolini, Estimates for Lebesgue constants in dimension two, Ann. Mat. Pura Appl. 156 (1990), 231-242. | Zbl 0778.42008
[002] [3] L. Brandolini, Fourier transform of characteristic functions and Lebesgue constants for multiple Fourier series, this volume, 51-59. | Zbl 0821.42009
[003] [4] A. Brondsted, An Introduction to Convex Polytopes, Springer, New York 1983.
[004] [5] M. Carenini and P. M. Soardi, Sharp estimates for Lebesgue constants, Proc. Amer. Math. Soc. 89 (1983), 449-452. | Zbl 0523.42009
[005] [6] D. I. Cartwright and P. M. Soardi, Best conditions for the norm convergence of Fourier series, J. Approx. Theory 38 (1983), 344-353. | Zbl 0516.42020
[006] [7] F. Cazzaniga and G. Travaglini, On pointwise convergence and localization for Fourier series on compact Lie groups, Arch. Math. (Basel), to appear. | Zbl 0779.43003
[007] [8] J.-L. Clerc, Sommes de Riesz et multiplicateurs sur un groupe de Lie compact, Ann. Inst. Fourier (Grenoble) 24 (1) (1974), 149-172. | Zbl 0273.22011
[008] [9] L. Colzani, S. Giulini, G. Travaglini and M. Vignati, Pointwise convergence of Fourier series on compact Lie groups, Colloq. Math. 60/61 (1990), 379-386. | Zbl 0751.43008
[009] [10] C. Fefferman, On the convergence of multiple Fourier series, Bull. Amer. Math. Soc. 77 (1971), 744-745. | Zbl 0234.42008
[010] [11] S. Giulini and G. Travaglini, Sharp estimates for Lebesgue constants on compact Lie groups, J. Funct. Anal. 68 (1986), 106-116. | Zbl 0664.22008
[011] [12] C. Herz, On the mean inversion of Fourier and Hankel transforms, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 996-999. | Zbl 0059.09901
[012] [13] J. M. Lopez and K. A. Ross, Sidon Sets, Marcel Dekker, New York 1975.
[013] [14] A. N. Podkorytov, Summation of multiple Fourier series over polyhedra, Vestnik Leningrad. Univ. Math. 13 (1981), 69-77. | Zbl 0477.42007
[014] [15] P. M. Soardi, Serie di Fourier in più variabili, U.M.I., Bologna 1984.
[015] [16] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton 1971. | Zbl 0232.42007
[016] [17] V. S. Varadarajan, Lie Groups, Lie Algebras and their Representations, Prentice-Hall, Englewood Cliffs 1974. | Zbl 0371.22001
[017] [18] V. A. Yudin, Behaviour of Lebesgue constants, Mat. Zametki 17 (1975), 401-405 (in Russian).
[018] [19] V. A. Yudin, Lower bound for Lebesgue constants, ibid. 25 (1979), 119-122 (in Russian).
[019] [20] A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge 1968. | Zbl 0157.38204