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On the order of approximation of a continuous 2π-periodic function by Fejer and Poisson means of its Fourier series

Abstract

Let σn (f) and Pr (f) be, respectively, the Fejer and Poisson means of the Fourier series of the functionf. The present work considers problems associated with the rapidity of approximation of a continuous 2π-periodic function by means of Fejer and Poisson processes, and gives, in particular, an upper bound to the deviation of the Fejer and Poisson processes from the function in terms of moduli of continuity, and a lower bound to ‖σn (f)−f‖ in terms of functionals composed of best approximations to the functionf; in addition, some relationships among the quantities ∥ Pr (f)−f ∥ and ‖σn (f)−f‖ are established.

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Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 21–32, July, 1968.

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Zhuk, V.V. On the order of approximation of a continuous 2π-periodic function by Fejer and Poisson means of its Fourier series. Mathematical Notes of the Academy of Sciences of the USSR 4, 500–508 (1968). https://doi.org/10.1007/BF01429810

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  • DOI: https://doi.org/10.1007/BF01429810

Keywords

  • Fourier
  • Fourier Series
  • Poisson Process
  • Poisson Means