On the approximation by trigonometric polynomials in weighted Lorentz spaces

Abstract

We obtain estimates of structural characteristics of 2π-periodic functions by the best trigonometric approximations in weighted Lorentz spaces, and show that the order of generalized modulus of smoothness depends not only on the rate of the best approximation, but also on the metric of the spaces. In weighted Lorentz spacesLps, this influence is expressed not only in terms of the parameterp, but also in terms of the second parameters.

Multivariate Approximation Theorems in Weighted Lorentz Spaces

“…In [8], these results were studied for nonweighted Lebesgue spaces. In one-variable weighted Lebesgue and Lorentz spaces, similar results were proved in [15,17,22]. …”

“…These results can be found in the monographs [7,21] for Lebesgue L p spaces. In the weighted Lebesgue and Lorentz spaces with Muckenhoupt A p -weights, some results were obtained in [1][2][3][10][11][12][13][14][15][16][17]22,23].…”

Improved direct and converse theorems in weighted Lorentz spaces

“…The constant C A p is called the Muckenhoupt constant of ω. By the proof of [14,Prop. 3.3], we know that…”

Jackson and Stechkin type inequalities of trigonometric approximation inAp;q()w;#

“…Variable exponent weighted Lebesgue spaces have important applications in the theory of elasticity, fluid dynamics, and differential equations [15,24]. Also, in weighted Lorentz spaces L p,q w (T) , which is a generalization of L p w (T) , direct and inverse theorems of approximation theory were proved in [8,9,21,28]. In this paper, we prove the direct and inverse theorems of trigonometric approximation in A p,q (•) w,ϑ (T) .…”