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Lebesgue constants of multiple de la Vallée-Poussin sums over polyhedra

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Abstract

One considers linear summation methods for the multiple Fourier series

the multidimensional analogues of the de la Vallé-Poussin sums. The summation of the Fourier series is carried out over the homotheties of an m-dimensional starshaped polyhedron Λ. It is shown that if Λ has rational vertices, then the Lebesgue constants of the considered methods, with the accuracy of O((p+1)−1. logm−1 (n+2)) are equal to

where

is the Fourier transform of the function ϕ. The exact value of the principal term of the Lebesgue constant is computed in two particular cases: 1) Λ is obtained from an m-dimensional cube by means of a linear nonsingular transformation; 2) ρ=0. Λ is an m-dimensional simplex.

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Literature cited

  1. A. F. Timan, Theory of Approximation of Functions of a Real Variable [in Russian], Fizmatgiz, Moscow (1960).

    Google Scholar 

  2. I. K. Daugavet, “On the Lebesgue constants for double Fourier series,” in; Methods of Computation, No. 6, Leningrad Univ. (1970), pp. 8–13.

    Google Scholar 

  3. O. I. Kuznetsova, “On certain properties of polynomial operators of triangular form in the space of continuous periodic functions of two variables,” Dokl. Akad. Nauk SSSR,223, No. 6, 1304–1306 (1975).

    Google Scholar 

  4. M. A. Skopina, “The Lebesgue constants for multiple Fourier sums over an oblique parallelepiped,” No. 2156-79, Deposited June 13, 1979.

  5. A. N. Podkorytov, “The summation of multiple Fourier series over polyhedra,” Vestn. Leningr. Gos. Univ., No. 1, Ser. Mat. Mekh. Astron., Issue 1, 51–58 (1980).

    Google Scholar 

  6. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press (1971).

  7. É. S. Belinskii, “An application of the Fourier transform to the summability of Fourier series,” Sib. Mat. Zh.,18, No. 3, 497–511 (1977).

    Google Scholar 

  8. É. S. Belinskii and I. R. Liflyand, “On the asymptotic behavior of the Lebesgue constants of radial summation methods,” in: Constructive Function Theory and the Theory of Mappings [in Russian], Naukova Dumka (1981), pp. 49–70.

  9. A. N. Podkorytov, “The order of growth of the Lebesgue constants of Fourier series over polyhedra,” Vestn. Leningr. Univ., No. 7, Ser. Mat. Mekh. Astron., Issue 2, 110–111 (1982).

    Google Scholar 

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Authors
  1. M. A. Skopina
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 154–165, 1983.

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Skopina, M.A. Lebesgue constants of multiple de la Vallée-Poussin sums over polyhedra. J Math Sci 26, 2404–2411 (1984). https://doi.org/10.1007/BF01680022

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  • DOI: https://doi.org/10.1007/BF01680022

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