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

Abstract

Abstract In this paper, we generalize and extend the Baskakov-Kantorovich operators by constructing the ( p , q ) $(p, q)$ -Baskakov Kantorovich operators ( 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 . $$ \begin{aligned} (\Upsilon _{n,b,p,q} h) (x) = [ n ]_{p,q} \sum_{b=0}^{ \infty}q^{b-1} \upsilon _{b,n}^{p,q}(x) \int _{\mathbb{R}}h(y)\Psi \biggl( [ n ] _{p,q} \frac{q^{b-1}}{p^{n-1}}y - [ b ] _{p,q} \biggr) \,d_{p,q}y. \end{aligned} $$ 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 ≤ b ≤ ω $0 \leq b \leq \omega $ , such that ω ∈ N $\omega \in \mathbb{N}$ , Ψ ω $\Psi _{\omega}$ is a continuous derivative function, and 0 0 $\gamma > 0$ with the property Ψ ⊂ [ 0 , γ ] $\Psi \subset [ 0, \gamma ] $ , (b) its first ω moment vanishes, that is, for 1 ≤ b ≤ ω $1 \leq b \leq \omega $ , we have that ∫ R y b Ψ ( y ) d p , q y = 0 $\int _{\mathbb{R}}y^{b}\Psi (y)\,d_{p,q}y = 0$ , (c) and ∫ R Ψ ( y ) d p , q y = 1 $\int _{\mathbb{R}}\Psi (y)\,d_{p,q}y = 1$ . Furthermore, we estimate the moments and norm of the new operators. And finally, we give an upper bound for the operator’s norm.

Keywords

• Kantorovich q-Baskakov operators
• Classical ( p , q ) $(p,q)$ -Baskakov operators
• Modified Kantorovich ( p , q ) $(p,q)$ -Baskakov operators
• ( p , q ) $(p, q)$ -integer
• ( p , q ) $(p, q)$ -power basis
• ( p , q ) $(p, q)$ -derivative