Approximation in weighted Orlicz spaces

Abstract

ABSTRACT. We prove some direct and converse theorems of trigonometric approximation in weighted Orlicz spaces with weights satisfying so called Muckenhoupt's A p condition.

A Modulus of Smoothness for Some Banach Function Spaces

“…Considering Example 5 of §2, clearly (1.3) improves the inequality (1.4) for r ≥ 2. Similarly (1.3) also improves direct theorems obtained in [3,4,5,17,18] for r ≥ 2.…”

“…So the Orlicz norm (2.4) satisfies the property (I'). If the conditions ϕ α quasiconvex for some α ∈ (0, 1), ϕ ∈ ∆ 2 and ω ∈ A p(ϕ) are fulfilled, then the properties (II)-(VI) were proved in [5]. Let M be a N -function.…”

Some inequalities of trigonometric approximation in weighted Orlicz spaces

“…For the more general Orlicz spaces, similar problems are investigated for Muckenhoupt weights. The direct and inverse theorems of trigonometric approximation (with Gadjieva modulus) in Orlicz spaces with Muckenhoupt weights were proved in [IG] (weight is inside and M is convex Young function) and [AI1] (weight is outside and M is quasiconvex Young function). Also some inverse theorems of trigonometric approximation of functions and its fractional derivatives in Orlicz spaces with Muckenhoupt weights were proved in [AI].…”

Approximation of periodic functions in weighted Orlicz spaces

“…This modulus of smoothness is well defined, because σ h is a bounded linear operator on L ϕ,ω (T) under the conditions that ϕ ∈ ∆ 2 , ϕ α is quasiconvex for some α, 0 < α < 1, and ω ∈ A p(ϕ) ( [1]). We define the shift operator σ h and the modulus of smoothness Ω r ϕ,ω in this way, because the space L ϕ,ω (T) is not, in general, invariant under the usual shift f (x) → f (x + h).…”