Direct theorems of trigonometric approximation for variable exponent Lebesgue spaces

• Ramazan Akgün ^[1]
  1. [1] Balıkesir University

     Balıkesir University

     Turquía

     
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 60, Nº. 1, 2019, págs. 121-135
• Idioma: inglés
• DOI: 10.33044/revuma.v60n1a08
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• Resumen

  • Jackson type direct theorems are considered in variable exponent Lebesgue spaces Lp(x) with exponent p(x) satisfying 1≤esinfx∈0,2π]p(x), esssup∈[0,2π]p(x)<∞, and the Dini–Lipschitz condition. Jackson type direct inequalities of trigonometric approximation are obtained for the modulus of smoothness based on one sided Steklov averages Zvf(⋅):=1v∫v0f(⋅+t)dt in these spaces. We give the main properties of the modulus of smoothness Ωr(f,v)p(⋅):=∥(I−Zv)rf∥p(⋅)(∈N)in Lp(x), where I is the identity operator. An equivalence of the modulus of smoothness and Peetre's K-functional is established.