Convergence of linking Baskakov-type operators

In this paper we consider a link Bn,ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{n,\rho }$$\end{document} between Baskakov type operators Bn,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{n,\infty }$$\end{document} and genuine Baskakov–Durrmeyer type operators Bn,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ B_{n,1}$$\end{document} depending on a positive real parameter ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}. The topic of the present paper is the pointwise limit relation Bn,ρfx→Bn,∞fx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( B_{n,\rho }f\right) \left( x\right) \rightarrow \left( B_{n,\infty }f\right) \left( x\right) $$\end{document} as ρ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho \rightarrow \infty $$\end{document} for x≥0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ge 0.$$\end{document} As a main result we derive uniform convergence on each compact subinterval of the positive real axis for all continuous functions f of polynomial growth.


Introduction
The so-called Baskakov type operators depending on a parameter c ∈ R were defined by Baskakov [1]. They are suitable for the approximation of functions being continuous on the underlying interval. It is well known that for the special choice of c = − 1, c = 0 and c = 1, respectively, one derives the classical Bernstein, Szász-Mirakjan and Baskakov operators, respectively.
In the context of approximation of integrable functions the Baskakov-Durrmeyer type operators play an important role. The construction is based on the corresponding definition of Bernstein-Durrmeyer operators (see [3,6,8,9,14] for further definitions). Although they have outstanding properties, i.e., they commute and are self-adjoint, they don't interpolate at finite endpoints of the interval and they only preserve constants. If the functions have supplemental finite limits at finite endpoints of the interval a modification of the Baskakov-Durrmeyer type operators leads to the genuine variants which, like the classical operators, preserve all linear functions and interpolate at finite endpoints of the interval.
During the last years the question arose how the classical operators and their genuine counterparts are connected. This investigation was initiated by Pȃltȃnea [10,11] by introducing a non-trivial link between Bernstein and Szász-Mirakjan operators, respectively, and their genuine Durrmeyer modifications (see also [4,5,12]).
The topic of the present paper is the investigation of convergence properties of linking Baskakov-Durrmeyer type operators for c > 0 when the linking parameter ρ tends to infinity.
Let c, n, ν, ρ ∈ R, such that n > c ≥ 0, ρ > 0, ν ≥ 0, and x ∈ [0, ∞). Then the basis functions are given by It can be shown that lim c→0 For the sake of brevity and in order to be consistent with notations in several references, we put throughout the paper for ν ≥ 1/ρ. For c > 0, we have the explicit representation In the following definition we assume that f : [0, ∞) → R is given in such a way that the corresponding integrals and series are convergent.
and the genuine Baskakov-Durrmeyer type operators are denoted by Depending on a parameter ρ > 0 the linking operators are given by Note that the genuine Baskakov-Durrmeyer type operators (1.2) are usually defined in the more explicit form Setting c = 0 in (1.2) leads to the Phillips operators [13], the case c > 0 was investigated in [15]. To the best of our knowledge, the case c = 0 in (1.3) was first considered in [11] and the general case in [7].
The purpose of this paper is to prove the limit lim ρ→∞ for all continuous functions f on [0, ∞) of polynomial growth. In the following theorem we state our main result.
Further results concerning the limit of the operators B n,ρ as ρ → ∞ for c ≥ 0 were proved in [2,7]. From the explicit representations of the images of polynomials for all operators B n,ρ it was possible to derive immediately that for c ≥ 0 uniformly on every compact subinterval of [0, ∞) for each polynomial p (see [7, Theorem 1, Theorem 2, Corollary 1]). A different function space was considered in [2] for the case uniformly on every compact subinterval of [0, ∞) (see [2, Lemma 5, Corollary 3]).

Auxiliary results
In this section we present several lemmata which are needed for the proof of our main result in Sect. 3. Let W [0, ∞) be the class of all locally integrable functions on [0, ∞) of polynomial growth, which are bounded on each compact subinterval of [0, ∞). Obviously, for every c > 0, each function f of this class satisfies an estimate of the type | f (x)| ≤ M (1 + cx) q for x ≥ 0 with certain constants M, q > 0. We start with the following observation.

Lemma 2.1 Let I ⊂ [0, ∞) be a compact interval. For f
for all x ≥ 0 and y ∈ I .
x ≥ 0 and y ∈ I with a certain positive constant M.
The following lemma guarantees the convergence of

Proof
As Observing that for sufficiently large ρ > 0, leads to the desired estimate.
Since the integral has a finite value, we derive by applying the estimate in Lemma 2.3 for .

This entails
with a positive constant α if T is sufficiently large.
Proof Let 1 ≤ ν ≤ N . As in the above proof we have