The estimates of approximation by using a new type of weighted modulus of continuity

Abstract

In this paper, we introduce a new type modulus of continuity for function f belonging to a particular weighted subspace of C [0, ∞) and show that it has some properties of ordinary modulus of continuity. We obtain some estimates of approximation of functions with respect to a suitable weighted norm via the new type moduli of continuity. Finally, we give some examples.

Linear positive operators constructed by using Beta-type bases

“…Keeping the same notation, from this moment, we consider the weight ρ given byρ(x) = 1 + ax + bx 2 , x ≥ 0, (3.10)where a ≥ 1 and b > 0. Since ρ(0) = 1 and inf x≥0 ρ ′ (x) ≥ 1, in accordance with[5], we can use the weighted modulus Ω ρ (f ; •) defined byΩ ρ (f ; δ) = sup x,t∈R + |ρ(t)−ρ(x)|≤δ |f (t) − f (x)| (|ρ(t) − ρ(x)| + 1)ρ(x), δ > 0,(3.11) for each f ∈ C ρ (R + ). Among the properties of this modulus proved by Gadjiev and Aral[5, Lemmas 4,5] we recall lim δ→0 Ω ρ (f ; δ) = 0, (3.12)…”

The Approximation Szász-Chlodowsky Type Operators Involving Gould-Hopper Type Polynomials

“…Theorem 10 (see [12]). Let ( ) be the sequences of linear positive operators and ( ) ≤ ( ), = 1, 2, 3.…”

Approximation by Generalized Lupaş Operators Based on q-Integers

“…Now, we recall local approximation in terms of α order Lipschitz-type maximal functions given in [27]. Let ρ be a function satisfying the conditions (ρ 1 ), (ρ 2 ), 0 < α ≤ 1 and Lip M (ρ(u); α), M ≥ 0 is the set of functions f satisfying the inequality…”