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1993 | 65 | 1 | 103-116
Tytu艂 artyku艂u

Polyhedral summability of multiple Fourier series (and explicit formulas for Dirichlet kernels on $饾晪^n$ and on compact Lie groups)

Tre艣膰 / Zawarto艣膰
Warianty tytu艂u
J臋zyki publikacji
EN
Abstrakty
EN
We study polyhedral Dirichlet kernels on the n-dimensional torus and we write a fairly simple formula which extends the one-dimensional identity $鈭慱{j=-N}^N e^{ijt} = sin((N+(1/2))t) / sin((1/2)t)$. 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.
Rocznik
Tom
65
Numer
1
Strony
103-116
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-05-26
poprawiono
1992-11-26
Tw贸rcy
  • Dipartimento di Matematica, Politecnico di Torino, Corso Duca Degli Abruzzi, 24 10129 Torino, Italy
Bibliografia
  • [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.
  • [2] L. Brandolini, Estimates for Lebesgue constants in dimension two, Ann. Mat. Pura Appl. 156 (1990), 231-242.
  • [3] L. Brandolini, Fourier transform of characteristic functions and Lebesgue constants for multiple Fourier series, this volume, 51-59.
  • [4] A. Brondsted, An Introduction to Convex Polytopes, Springer, New York 1983.
  • [5] M. Carenini and P. M. Soardi, Sharp estimates for Lebesgue constants, Proc. Amer. Math. Soc. 89 (1983), 449-452.
  • [6] D. I. Cartwright and P. M. Soardi, Best conditions for the norm convergence of Fourier series, J. Approx. Theory 38 (1983), 344-353.
  • [7] F. Cazzaniga and G. Travaglini, On pointwise convergence and localization for Fourier series on compact Lie groups, Arch. Math. (Basel), to appear.
  • [8] J.-L. Clerc, Sommes de Riesz et multiplicateurs sur un groupe de Lie compact, Ann. Inst. Fourier (Grenoble) 24 (1) (1974), 149-172.
  • [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.
  • [10] C. Fefferman, On the convergence of multiple Fourier series, Bull. Amer. Math. Soc. 77 (1971), 744-745.
  • [11] S. Giulini and G. Travaglini, Sharp estimates for Lebesgue constants on compact Lie groups, J. Funct. Anal. 68 (1986), 106-116.
  • [12] C. Herz, On the mean inversion of Fourier and Hankel transforms, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 996-999.
  • [13] J. M. Lopez and K. A. Ross, Sidon Sets, Marcel Dekker, New York 1975.
  • [14] A. N. Podkorytov, Summation of multiple Fourier series over polyhedra, Vestnik Leningrad. Univ. Math. 13 (1981), 69-77.
  • [15] P. M. Soardi, Serie di Fourier in pi霉 variabili, U.M.I., Bologna 1984.
  • [16] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton 1971.
  • [17] V. S. Varadarajan, Lie Groups, Lie Algebras and their Representations, Prentice-Hall, Englewood Cliffs 1974.
  • [18] V. A. Yudin, Behaviour of Lebesgue constants, Mat. Zametki 17 (1975), 401-405 (in Russian).
  • [19] V. A. Yudin, Lower bound for Lebesgue constants, ibid. 25 (1979), 119-122 (in Russian).
  • [20] A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge 1968.
Typ dokumentu
Bibliografia
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