Whitney’s theorem for local anisotropic polynomial Lp-approximation, 0 < p < 1

• Dinh Dung ^[2] ; Nguyen Van Dung ^[1] ; Nguyen Dinh Hoa ^[2]
  1. [1] University of Transport and Communications

     University of Transport and Communications

     Vietnam

     
  2. [2] Vietnam National University
• Localización: Jaen journal on approximation, ISSN 1889-3066, ISSN-e 1989-7251, Vol. 6, Nº. 1, 2014, págs. 69-85
• Idioma: inglés
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• Resumen

  • Dinh Düng and T. Ullrich have proven a multivariate Whitney’s theorem for the local anisotropic polynomial approximation in Lp(Q) for 1 ≤ p ≤ ∞, where Q is a d-parallelepiped in Rᵈ with sides parallel to the coordinate axes. They considered the error of best approximation of a function f by algebraic polynomials of fixed degree at most ri− 1 in the variable xi, i = 1, ..., d. The convergence rate of the approximation error when the size of Q goes to 0 is characterized by a so-called total mixed modulus of smoothness. The method of proof used by these authors is not suitable for the case 0 < p < 1. In the present paper, by a different technique we prove this theorem for 0 < p ≤ ∞.