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

Abstract

In this paper, we generalize and extend the Baskakov-Kantorovich operators by constructing the (p, q)-Baskakov Kantorovich operators (ϒn,b,p,qh)(x)=[n]p,q∑b=0∞qb−1υb,np,q(x)∫Rh(y)Ψ([n]p,qqb−1pn−1y−[b]p,q)dp,qy.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\begin{aligned} (\\Upsilon _{n,b,p,q} h) (x) = [ n ]_{p,q} \\sum_{b=0}^{ \\infty}q^{b-1} \\upsilon _{b,n}^{p,q}(x) \\int _{\\mathbb{R}}h(y)\\Psi \\biggl( [ n ] _{p,q} \\frac{q^{b-1}}{p^{n-1}}y - [ b ] _{p,q} \\biggr) \\,d_{p,q}y. \\end{aligned} $$\\end{document} The modified Kantorovich (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 , such that omega in mathbb{N}, Psi _{omega} is a continuous derivative function, and 0< q< p leq 1, we have int _{mathbb{R}}x^{b}Psi _{omega}(x),d_{p,q}x = 0 . Also, for every Psi in L_{infty},there exists a finite constant γ such that gamma > 0 with the property Psi subset [ 0, gamma ] ,its first ω moment vanishes, that is, for 1 leq b leq omega , we have that int _{mathbb{R}}y^{b}Psi (y),d_{p,q}y = 0,and 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.