Approximation in weighted Lorentz spaces On moduli of smoothness and approximation by trigonometric polynomials in weighted Lorentz spaces

Abstract

We investigate the approximation properties of the functions by trigonometric polynomials in weighted Lorentz spaces with weights satisfying so called Muckenhoupt's Ap condition. Relations between moduli of smoothness of the derivatives of the functions and those of the functions itself are studied. In weighted Lorentz spaces we also prove a theorem on the relationship between the derivatives of a polynomial of best approximation and the best approximation of the function. Moreover, we study relationship between modulus of smoothness of the function and its de la Vallée-Poussin sums in these spaces.

Gadjieva's conjecture, K -functionals, and some applications in weighted Lebesgue spaces

“…≲ Ω r (f, π/n) p,γ ≲ Ω r (f, 1/n) p,γ and hence R 2r (f, 1/n, p, γ) ≲ Ω r (f, 1/n) p,γ . For the reverse inequality we use (23) and Lemma 15 (with h = 1/n ):…”