For citation:
Sharapudinov I. I. Approximation of Smooth Functions in Lp(x)2π by Vallee–Poussin Means. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2013, vol. 13, iss. 1, pp. 45-49. DOI: 10.18500/1816-9791-2013-13-1-1-45-49, EDN: SMXXIP
Approximation of Smooth Functions in Lp(x)2π by Vallee–Poussin Means
Variable exponent p(x) Lebesgue spaces Lp(x)2π is considered. For f ∈ Lp(x)2π Vallee–Poussin means Vnm(f, x) can be defined as Vnm(f, x) = 1/(m+1)Ʃl=0mSn+l(f, x), where Sk(f, x) –- partial Fourier sum of f(x) of order k. Approximative properties of operators Vnm(f) = Vnm(f, x) are investigated in Lp(x)2π. Let p(x) ≥ 1 be 2π-periodical variable exponent that satisfies Dini–Lipschitz condition. When m = n − 1 and m = n the following estimate is proved: ||f − Vnm(f)||p(·) ≤ (cr(p)/nr )En(f(r))p(·) where En(f(r))p(·) is the best approximation of function f(r)(x) by trigonometric polynomials of order n in Lp(x)2π.
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