On the Moduli of Smoothness with Jacobi Weights

Abstract

The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights (1 − x) α (1 + x) β for functions in the Jacobi weighted L p [−1, 1], 0 < p ≤ ∞, spaces. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted L p spaces. If 1 ≤ p ≤ ∞, then these moduli are equivalent to certain weighted K-functionals (and so they are equivalent to certain weighted Ditzian-Totik moduli of smoothness for these p),

Simultaneous approximation by Bernstein polynomials with integer coefficients

“…). Throughout c denotes positive constants, whose value is independent of f and n. Instead of ω 2 ϕ (f, t) we can use the moduli defined and considered in [11,12], [10,15,16,17,18,19,22], or [9].…”

Equivalence of Weighted Dt-Moduli of Convex Functions

“…Hierarchy foundations of the moduli of smoothness are begun modern with the work of Ditzian and Totik (1987), (see [6]), and Kopotun (2006Kopotun ( -2019, (see [8,9,10,11,12,14,15,16,18]). Ditzian and Totik established better continuous moduli of the function in a normed space, then Kopotun contributed to properties of various moduli of smoothness like univariate piecewise polynomial functions (splines) [16].…”

On Some Properties of Moduli of Smoothness with Jacobi Weights

“…We denote by P n the set of all algebraic polynomials of degree ≤ n − 1, and L where AC loc denotes the set of functions which are locally absolutely continuous in (−1, 1), and ϕ(x) := √ 1 − x 2 . Also (see [5]), for k, r ∈ N and f ∈ B r p (w α,β ), let ω ϕ k,r (f (r) , t) α,β,p := sup 0≤h≤t W r/2+α,r/2+β kh…”

“…i.e., Ψ ϕ k,r is "the main part modulus Ω ϕ k,r with A = 0". However, we want to emphasize that while Ω ϕ k,r (f (r) , A, t) α,β,p with A > 0 and ω ϕ k,r (f (r) , t) α,β,p are bounded for all f ∈ B r p (w α,β ) (see [5,Lemma 2.4]), modulus Ψ ϕ k,r (f (r) , t) α,β,p may be infinite for such functions (for example, this is the case for f such that f (r) (x) = (1 − x) −γ with 1/p ≤ γ < α + r/2 + 1/p). Remark 1.1.…”

“…We denote by P n the set of all algebraic polynomials of degree ≤ n − 1, and L Following [5] we denote B 0 p (w α,β ) := L α,β p , and…”