Onq‐analogue of a parametric generalization of Baskakov operators

Abstract

The purpose of this paper is to define a q‐analogue of a parametric generalization of Baskakov operators introduced by Aral and Erbay (Math. Commun. 24(1) (2019), 119‐131). We establish some local direct results for these operators by means of the modulus of continuity and the Peetre's K‐functional. Weighted approximation properties are also established. Next, we construct a bivariate case of the above q‐operators and determine the rate of convergence in terms of the moduli of continuity. A Voronovskaja‐type theorem is derived too. Further, we study the rate of approximation of Bögel continuous and Bögel differentiable functions by the associated generalized Boolean sum (GBS) operators with the aid of mixed modulus of smoothness. We illustrate the rate of approximation of the q‐operators, their bivariate, and the GBS cases by means of graphics and tables and show that the GBS operators provide better rate of convergence than the bivariate operators.

A new kind of Bi-variate $ \lambda $ -Bernstein-Kantorovich type operator with shifted knots and its associated GBS form

“…m1,m2,λ1,λ2 (|f (t, y) − f (x, y)| ; x, y) , using the definition of partial moduli of continuity with respect to x and y as defined in (9), we have K α1,β1,α2,β2 m1,m2,λ1,λ2 (f (t, s); x, y) − f (x, y) ≤ K α1,β1,α2,β2 m1,m2,λ1,λ2 (ω (2) (f ; |s − y|); x, y) +K α1,β1,α2,β2 m1,m2,λ1,λ2 (ω (1) (f ; |t − x|); x, y) . From the property of modulus of continuity and using the Cauchy-Schwarz inequality, we can write for δ 1 , δ 2 > 0…”

“…For some more related and recent works in this direction, we refer the reader to (cf. [5], [2], [13], [14], [16], [17], [15], [22], and [25] etc. ).…”

“…x ∈ [ α m+β , m+α m+β ] and b (α,β) m,j (x)(j = 0, 1, ...m) are as given by (2). We note that as m → ∞, the interval [ α m+β , m+α m+β ] → [0, 1] and whenever α = β = 0 and λ = 0, the basis functions b ∧(α,β) m,j (λ, x) take the form of the classical Bernstein functions.…”