Realization and characterization of modulus of smoothness in weighted Lebesgue spaces

Abstract

We obtain a characterization of modulus of smoothnes of fractional order in the Lebesgue spaces L p ω , 1 < p < ∞, with weights ω satisfying the Muckenhoupt's A p condition. Also, a realization result and equivalence between modulus of smoothness and the Peetre K-functional are proved in L p ω for 1 < p < ∞ and ω ∈ A p. Theorem 1 ([31]). Suppose that r > 0 and 1 ≤ p ≤ ∞. In this case (a) If f ∈ L p , then there exists a function ϕ ∈ Φ r such that ϕ (t) ≈ ω r (f, t) p holds for any t ∈ (0, ∞), where equivalence constants are depend only on r and p. (b) If ϕ ∈ Φ r , then there exist f ∈ L p and a positive real number t 0 such that ω r (f, δ) p ≈ ϕ (δ) holds for any δ ∈ (0, t 0 ], where the equivalence constants are depend only on r and p.

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

“…As a corollary we can obtain a Jackson-Stechkin type inequality, which improves (for r ≥ 2) the Jackson-Stechkin type inequalities obtained in [2,3,16,20,29,30,43].…”

“…Indeed, (i) for Theorem 4 see Proposition 1 of [2]. For other theorems see the results given in [3].…”

Generalized Jacobian Weighted Modulus of Smoothness

“…The most important application of all moduli is their rule in approximating functions, especially by polynomials or/and neural networks, for more information, see [1]- [10]. It is not clear when the first weighted moduli of smoothness were introduced, but it is probably defined first by Ditizian and Totik in their book [1], they were working on linking the weighted moduli of smoothness to the weighted approximation [2] in weighted spaces such as [3]. The idea began with defining the weighted norm in [4] and it was followed by the investigating other weights for spaces and moduli.…”

On moduli of smoothness and averaged differences of fractional order

“…The modulus ω β (f, h) p has been introduced and studied in [21], see also [22] and [24,Ch.8], in which it is also called the linearized modulus of smoothness. Some applications of the modulus ω β (f, h) p as well as some of its modifications can be found in [1], [6], [9], [10], and [23].…”