On wavelets Kantorovich $(p,q)$-Baskakov operators and approximation properties

Abstract

AbstractIn this paper, we generalize and extend the Baskakov-Kantorovich operators by constructing the $(p, q)$ ( p , q ) -Baskakov Kantorovich operators $$ \begin{aligned} (\Upsilon _{n,b,p,q} h) (x) = [ n ]_{p,q} \sum_{b=0}^{ \infty}q^{b-1} \upsilon _{b,n}^{p,q}(x) \int _{\mathbb{R}}h(y)\Psi \biggl( [ n ] _{p,q} \frac{q^{b-1}}{p^{n-1}}y - [ b ] _{p,q} \biggr) \,d_{p,q}y. \end{aligned} $$ ( ϒ n , b , p , q h ) ( x ) = [ n ] p , q ∑ b = 0 ∞ q b − 1 υ b , n p , q ( x ) ∫ R h ( y ) Ψ ( [ n ] p , q q b − 1 p n − 1 y − [ b ] p , q ) d p , q y . The modified Kantorovich $(p, q)$ ( p , q ) -Baskakov operators do not generalize the Kantorovich q-Baskakov operators. Thus, we introduce a new form of this operator. We also introduce the following useful conditions, that is, for any $0 \leq b \leq \omega $ 0 ≤ b ≤ ω , such that $\omega \in \mathbb{N}$ ω ∈ N , $\Psi _{\omega}$ Ψ ω is a continuous derivative function, and $0< q< p \leq 1$ 0 < q < p ≤ 1 , we have $\int _{\mathbb{R}}x^{b}\Psi _{\omega}(x)\,d_{p,q}x = 0 $ ∫ R x b Ψ ω ( x ) d p , q x = 0 . Also, for every $\Psi \in L_{\infty}$ Ψ ∈ L ∞ , there exists a finite constant γ such that $\gamma > 0$ γ > 0 with the property $\Psi \subset [ 0, \gamma ] $ Ψ ⊂ [ 0 , γ ] , its first ω moment vanishes, that is, for $1 \leq b \leq \omega $ 1 ≤ b ≤ ω , we have that $\int _{\mathbb{R}}y^{b}\Psi (y)\,d_{p,q}y = 0$ ∫ R y b Ψ ( y ) d p , q y = 0 , and $\int _{\mathbb{R}}\Psi (y)\,d_{p,q}y = 1$ ∫ R Ψ ( y ) d p , q y = 1 . Furthermore, we estimate the moments and norm of the new operators. And finally, we give an upper bound for the operator’s norm.