Approximation by generalized Baskakov-beta operators and its convergence properties

The main intent of this paper is to innovate summation-integral type generalized Baskakov operators with weights of Beta basis functions whose construction depends on φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document} such that φ(0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (0) = 0$$\end{document} and infϰ∈[0,∞)φ′(ϰ)≥1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\inf \nolimits _{\varkappa \in [0,\infty )}\varphi ^{'}(\varkappa )\ge 1.$$\end{document} For these operators we compute rate of convergence by weighted modulus of continuity, pointwise convergence by proving Voronovskaya type theorem. Finally, we figure out quantitative estimates for the local approximation and provide some computer graphics.