Jackson-Stechkin type inequality in weighted Lorentz spaces

Abstract

Abstract. In the present work we consider the modulus of smoothness, defined by means of the Steklov operator in weighted Lorentz spaces and prove the Jackson-Stechkin type direct theorem of trigonometric approximation. In the particular case we obtain a result on the constructive characterization of the generalized Lipschitz classes defined in these spaces. Simultaneous approximation of functions is also considered.Mathematics subject classification (2010): 41A25, 41A27, 42A10.

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) .…”

A study on convolutions in weighted lorentz spaces using steklov means

“…The basic properties of this type of convolution operator used in the structure of the approximation polynomials and modulus of smoothness were investigated in variable exponent Lebesgue space by Israfilov and Yirtici in [7] and weighted Lorentz spaces in [6,[8][9][10][11]. In addition, properties of convolution type transforms were also examined in [12][13][14][15].…”

Convolution and Jackson inequalities in Musielak–Orlicz spaces

“…In some weighted Banach function spaces, similar problems were studied in [6,7,9,17,18,30,34,36,37]. In general, Musielak-Orlicz spaces may not attain the translation invariance property, as can be seen in the case of variable exponent Lebesgue spaces L p (x) .…”