# Approximation with arbitrary order by certain linear positive operators

## Abstract

This paper aims to highlight classes of linear positive operators of discrete and integral type for which the rates in approximation of continuous functions and in quantitative estimates in Voronovskaya type results are of an arbitrarily small order. The operators act on functions defined on unbounded intervals and we achieve the intended purpose by using a strictly decreasing positive sequence $$(\lambda _{n})_{n\geq1}$$ such that $$lim_{n\rightarrow\infty}$$ $$\lambda_{n}=0$$, how fast we want. Particular cases are presented.

## Authors

Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

## Keywords

Linear and positive operator · Korovkin theorem · Voronovskaya type theorems · Modulus of continuity · Order of approximation

## Paper coordinates

O. Agratini, Approximation with arbitrary order by certain linear positive operators, Positivity, 22 (2018), pp. 1241-1254, https://doi.org/10.1007/s11117-018-0570-9

Positivity

Springer

1385-1292

1572-9281

## Paper (preprint) in HTML form

Approximation with arbitrary order by certain linear positive operators

# Approximation with arbitrary order by certain linear positive operators

Octavian Agratini and Sorin G. Gal Babeş-Bolyai University
Faculty of Mathematics and Computer Science
Str. Kogălniceanu 1
400084 Cluj-Napoca, Romania
Department of Mathematics and Computer Science
Str. Universităţii 1
###### Abstract.

This paper aims to highlight classes of linear positive operators of discrete and integral type for which the rates in approximation of continuous functions and in quantitative estimates in Voronovskaya type results are of an arbitrarily small order. The operators act on functions defined on unbounded intervals and we achieve the intended purpose by using a strictly decreasing positive sequence $(\lambda_{n})_{n\geq 1}$ such that $\lim\limits_{n\to\infty}\lambda_{n}=0$, how fast we want. Particular cases are presented.

Keywords and phrases: Linear and positive operator, Korovkin theorem, Voronovskaya type theorems, modulus of continuity, order of approximation.

Mathematics Subject Classification: 41A36, 41A25.

## 1. Introduction

A special branch of Approximation Theory is the approximation of functions by using linear sequences of operators, say $(L_{n})_{n\geq 1}$, the essential feature being that of positivity. The cornerstone of this approach is Bohman-Korovkin criterion [5], [15]. This criterion says: if $(L_{n}e_{k})_{n\geq 1}$ converges to $e_{k}$ uniformly on $[a,b]$, $k\in\{0,1,2\}$, for the test functions $e_{0}(x)=1$, $e_{1}(x)=x$, $e_{2}(x)=x^{2}$, then $(L_{n}f)_{n\geq 1}$ converges to $f$ uniformly on $[a,b]$ for each $f\in C([a,b])$. Usually, for continuous functions defined on an unbounded interval only pointwise convergence occurs. It is one of the reasons why the study of such operators is more productive.

We recall that two types of positive approximation processes are used – the discrete, respectively continuous form. In what follows we consider as a starting point a linear positive process of discrete type which approximate functions defined on unbounded intervals. Since a linear substitution maps the interior of any unbounded interval $J$ onto $(0,\infty)$ or $\mathbb{R}$, we have only two options. In our study we consider the benchmark interval $J=[0,\infty)=\mathbb{R}_{+}$ as it exhibits the problems caused by a finite endpoint and by the nonboundedness of the interval. Operators in questions are often designed as follows

 $(L_{n}f)(x)=\sum_{k=0}^{\infty}a_{k}(n;x)f(x_{n,k}),\ n\in\mathbb{N},\ x\in% \mathbb{R}_{+},$ (1.1)

where $a_{k}(n;\cdot):\mathbb{R}_{+}\to\mathbb{R}_{+}$ are continuous functions for each $n\in\mathbb{N}$, $k\in\{0\}\cup\mathbb{N}=\mathbb{N}_{0}$, and $\Delta_{n}=(x_{n,k})_{k\geq 0}$ is a net on $\mathbb{R}_{+}$. Here $f$ belongs to a function space for which the right-hand side of the relation (1.1) is well defined.

The most common case for classical operators is the equidistant network $x_{n,k}=k/n$, $k\geq 0$. We mention that in (1.1) could be a finite sum. For example, if we take

 $a_{k}(n;x)=\binom{n}{k}(1+x)^{-n}x^{k}\mbox{ for }0\leq k\leq n,\ a_{k}(n;x)=0% \mbox{ for }k>n,$

and we choose the net $\Delta_{n}=(k/(n-k+1))_{k=\overline{0,n}}$, then $L_{n}$, $n\in\mathbb{N}$, become Bleiman-Butzer-Hahn operators [4].

The disadvantage of the positive linear approximation processes is definitely determined by the fact that they have a low rate of convergence. For some spaces of functions considered, the error of approximation $|(L_{n}f)(x)-f(x)|$ is assessed by using the modulus of continuity $\omega$ of $f:I\to\mathbb{R}$ with argument $\delta$ defined by

 $\omega(f;\delta)=\sup\{|f(x)-f(y)|;x,y\in I,|x-y|\leq\delta\},\delta\geq 0.$ (1.2)

We are considering functions for which the supremum is finite, for example bounded functions, uniformly continuous functions, Lipschitz functions.

For a wide majority of linear positive operators, the error of approximation is described by using $\omega\left(f;1/\sqrt{n}\right)$ [8, page 40], consequently the order of approximation is ${\mathcal{O}}(n^{-1/2})$. Is it possible to construct classes of operators having a faster rate of convergence? Recently, such classes of operators acting on certain function spaces have been obtained.

Starting from (1.1) we consider a general class of discrete operators with the property that the order of approximation is improved, meaning that becomes arbitrarily small. Also, we prove that the same phenomenon holds for quantitative estimates in Voronovskaya type theorems. The next section consists in associating to this new process an integral generalization of Kantorovich type, obtaining for it the same kind of results. Particular sequences are identified in our construction, such that the result established can be applied to several cases.

## 2. A discrete class of operators

Let $\lambda=(\lambda_{n})_{n\geq 1}$ be a strictly decreasing sequence of positive numbers with the property

 $\lim\limits_{n\to\infty}\lambda_{n}=0.$ (2.1)

We modify the operators defined by (1.1) replacing $n$ with $\lambda_{n}^{-1}$. Also we consider the equidistant network having the nodes $x_{n,k}=k\lambda_{n}$. The discrete class of operators is given as follows

 $(L_{n}^{\langle\lambda\rangle}f)(x)=\sum_{k=0}^{\infty}a_{k}(\lambda_{n};x)f(k% \lambda_{n}),\ n\in\mathbb{N},\ x\geq 0,$ (2.2)

where $a_{k}(\lambda_{n};\cdot)$ are non-negative functions belonging to $C(\mathbb{R}_{+})$ and

 $f\in{\mathcal{D}}:=\{g\in C(\mathbb{R}_{+}):\mbox{the series in}\ (\ref{4})\ % \mbox{is absolutely convergent}\}.$ (2.3)

We are working under assumption that the monomials $e_{j}$, $0\leq j\leq 2$, belong to the above domain.

Clearly, $L_{n}^{\langle\lambda\rangle}$, $n\in\mathbb{N}$, are linear and positive operators. At this point we introduce the first and the second central moments of $L_{n}^{\langle\lambda\rangle}$ operators, i.e.,

 ${\mathcal{M}}_{i}(L_{n}^{\langle\lambda\rangle};x)=(L_{n}^{\langle\lambda% \rangle}\varphi_{x}^{i})(x),\mbox{ where }\varphi_{x}(t)=t-x,\ (t,x)\in\mathbb% {R}_{+}\times\mathbb{R}_{+},$

and $i\in\{1,2\}$. Concerning the operators defined by (2.2), we consider that for each $x\in\mathbb{R}_{+}$ and $n\in\mathbb{N}$ they satisfy the following properties

 $\sum_{k=0}^{\infty}a_{k}(\lambda_{n};x)=1,$ (2.4)
 ${\mathcal{M}}_{1}(L_{n}^{\langle\lambda\rangle};x)=\lambda_{n}l_{1}(x),$ (2.5)
 ${\mathcal{M}}_{2}(L_{n}^{\langle\lambda\rangle};x)=\lambda_{n}l_{2}(x)+\lambda% _{n}^{2}l_{3}(x),$ (2.6)

where $l_{j}\in C(\mathbb{R}_{+})$, $j\in\{1,2,3\}$.

Taking into account Bohman-Korovkin theorem, $(L_{n}^{\langle\lambda\rangle})_{n\geq 1}$ is an approximation process on the space $C(K)$, endowed with the uniform norm, $K\subseteq\mathbb{R}_{+}$ compact, if

 $\lim\limits_{n\to\infty}L_{n}^{\langle\lambda\rangle}e_{j}=e_{j},\ 0\leq j\leq 2,$

uniformly on $K$.

Based on the requirements (2.4), (2.5), (2.6), the above limits are achieved. On the other hand, our hypothesis package means much more. The real problem that arises here is whether there are classes of $(L_{n}^{\langle\lambda\rangle})_{n\geq 1}$ operators which meet all assumptions. We certify this fact presenting two examples. We also indicate successive steps taken to obtain them.

###### Example 2.1.

Ismail [13] introduced a generalization of Szász operators given by

 $(T_{n}f)(x)=\displaystyle\frac{e^{-nxH(1)}}{A(1)}\sum_{k=0}^{\infty}p_{k}(nx)f% \left(\displaystyle\frac{k}{n}\right),$ (2.7)

where $p_{k}$ are the Sheffer polynomials defined by

 $A(t)e^{xH(t)}=\sum_{k=0}^{\infty}p_{k}(x)t^{k},\ x\geq 0,\ |t|

with

 $A(z)=\sum_{k=0}^{\infty}c_{k}z^{k},\ H(z)=\sum_{k=1}^{\infty}h_{k}z^{k},$ (2.8)

analytic functions in a disk $|z|, $R>1$, such that $A(1)\neq 0$, $H^{\prime}(1)=1$, $c_{k}\in\mathbb{R}$ for $k\in\mathbb{N}_{0}$, $h_{k}\in\mathbb{R}$ for $k\in\mathbb{N}$, $c_{0}h_{1}\neq 0$ and supposing that $p_{k}(x)\geq 0$ for all $(k,x)\in\mathbb{N}_{0}\times\mathbb{R}_{+}$.

By using two sequences of non-negative real numbers $(\alpha_{n})_{n}$, $(\beta_{n})_{n}$ with the property that $(\beta_{n}/\alpha_{n})_{n}$ is strictly decreasing having null limit, Cetin and Ispir [6] introduced and studied a notable generalization of Szász operators attached to analytic functions $f$ of exponential growth in a compact disk of the complex plane, $|z|,

 $S_{n}(f;\alpha_{n},\beta_{n})(z)=e^{-\alpha_{n}z/\beta_{n}}\sum_{k=0}^{\infty}% \displaystyle\frac{1}{k!}\left(\displaystyle\frac{\alpha_{n}z}{\beta_{n}}% \right)^{k}f\left(\displaystyle\frac{k\beta_{n}}{\alpha_{n}}\right).$

For the particular case $z\in\mathbb{R}_{+}$, $\alpha_{n}=n$, $\beta_{n}=1$, these operators turn into genuine Szász operators [20].

In [10] Gal considered the Ismail’s kind generalization (2.7), this time the operators being applied to certain real-valued functions defined on $\mathbb{R}_{+}$. More precisely, the following operators are studied

 $T_{n}^{*}(f;\alpha_{n},\beta_{n})(x)=\displaystyle\frac{e^{-\alpha_{n}xH(1)/% \beta_{n}}}{A(1)}\sum_{k=0}^{\infty}p_{k}\left(\displaystyle\frac{\alpha_{n}x}% {\beta_{n}}\right)f\left(\displaystyle\frac{k\beta_{n}}{\alpha_{n}}\right),$ (2.9)

$x\in\mathbb{R}_{+}$, under the hypothesis on $A,H$ and $p_{k}$ specified at (2.8).

Setting $\lambda_{n}:=\beta_{n}/\alpha_{n}$, condition (2.1) is satisfied and we observe that these operators are of the form (2.2) with

 $a_{k}(\lambda_{n};x)=\displaystyle\frac{e^{-xH(1)/\lambda_{n}}}{A(1)}p_{k}(x/% \lambda_{n}),\ x\geq 0.$

For $\lambda_{n}=1/n$ we reobtain $T_{n}$ operators which have the form indicated in (1.1). Following [10, Lemma 2.1], the author proved

 $\displaystyle T_{n}^{*}(e_{0};\alpha_{n},\beta_{n})=1,$ $\displaystyle{\mathcal{M}}_{1}(T_{n}^{*};x)=\displaystyle\frac{\beta_{n}}{% \alpha_{n}}\,\displaystyle\frac{A^{\prime}(1)}{A(1)},$ $\displaystyle{\mathcal{M}}_{2}(T_{n}^{*};x)=x\displaystyle\frac{\beta_{n}}{% \alpha_{n}}(H^{\prime\prime}(1)+1)+\displaystyle\frac{\beta_{n}^{2}}{\alpha_{n% }^{2}}\,\displaystyle\frac{A^{\prime}(1)+A^{\prime\prime}(1)}{A(1)}.$

Comparing these identities to those describing by relations (2.4), (2.5), (2.6), we deduce

 $l_{1}(x)=\displaystyle\frac{A^{\prime}(1)}{A(1)},\ l_{2}(x)=(H^{\prime\prime}(% 1)+1)x,\ l_{3}(x)=\displaystyle\frac{A^{\prime}(1)+A^{\prime\prime}(1)}{A(1)},% \ x\geq 0.$
###### Example 2.2.

Set

 $E_{2}(\mathbb{R}_{+})=\{f\in C(\mathbb{R}_{+}):\ f(x)/(1+x^{2})\mbox{ is % convergent as }x\to\infty\}.$

Firstly we present Mastroianni operators [17]. Let $\varphi_{n}$, $n\in\mathbb{N}$, be real functions on $\mathbb{R}_{+}$ which are infinitely differentiable and strictly monotone on $\mathbb{R}_{+}$, satisfying the following properties:

$(P_{1})$ $\varphi_{n}(0)=1,\ n\in\mathbb{N}$;

$(P_{2})$ $\varphi_{n}$, $n\in\mathbb{N}$ are completely monotone on $\mathbb{R}_{+}$, i.e. $(-1)^{k}\varphi_{n}^{(k)}(x)\geq 0$ for each $x\in\mathbb{R}_{+}$ and $k\in\mathbb{N}_{0}$;

$(P_{3})$ for every $(n,k)\in\mathbb{N}\times\mathbb{N}_{0}$ there exists a positive integer $p(n,k)\in\mathbb{N}$ and a real function $\alpha_{n,k}:\mathbb{R}_{+}\to\mathbb{R}$ such that

 $\displaystyle\varphi_{n}^{(i+k)}(x)=(-1)^{k}\varphi_{p(n,k)}^{(i)}(x)\alpha_{n% ,k}(x),\ i\in\mathbb{N}_{0},\ x\in\mathbb{R}_{+},\mbox{ and}$ $\displaystyle\lim\limits_{n\to\infty}\displaystyle\frac{n}{p(n,k)}=\lim\limits% _{n\to\infty}\displaystyle\frac{\alpha_{n,k}(0)}{n^{k}}=1.$

To the sequence $(\varphi_{n})_{n\geq 1}$ Mastroianni associated the linear and positive operators $M_{n}$, $n\geq 1$, acting from $E_{2}(\mathbb{R}_{+})$ to $C(\mathbb{R}_{+})$ as follows

 $(M_{n}f)(x)=\sum_{k=0}^{\infty}\displaystyle\frac{(-1)^{k}}{k!}\varphi_{n}^{(k% )}(x)f\left(\displaystyle\frac{k}{n}\right),\ x\in\mathbb{R}_{+}.$ (2.10)

We consider $\varphi_{n}(x)=(1+x)^{-n}$, $x\geq 0$. These functions are infinitely differentiable and strictly decreasing on the domain. Moreover, $\varphi_{n}(0)=1$ and $\varphi_{n}^{(k)}(x)=(-1)^{k}n(n+1)\ldots(n+k-1)(1+x)^{-n-k}$, $k\geq 1$, consequently conditions $(P_{1})$ and $(P_{2})$ are fulfilled. We choose

 $p(n,k)=n+k\mbox{ and }\alpha_{n,k}(x)=n(n+1)\ldots(n+k-1)(1+x)^{-k}$

which implies $\varphi_{n}^{(i+k)}(x)=(-1)^{k}\varphi_{n+k}^{(i)}(x)\alpha_{n,k}(x)$. Therefore, property $(P_{3})$ also occurs. In this particular case, the operators defined by (2.10) turn into the celebrated Baskakov operators [3]

 $\displaystyle(V_{n}f)(x)$ $\displaystyle=(1+x)^{-n}f(0)$ $\displaystyle+(1+x)^{-n}\sum_{k=1}^{\infty}\displaystyle\frac{n(n+1)\ldots(n+k% -1)}{k!}\left(\displaystyle\frac{x}{1+x}\right)^{k}f\left(\displaystyle\frac{k% }{n}\right).$

Recently, Gal and Opriş [11] introduced a modified Baskakov type operator. In the definition of $\varphi_{n}(x)=(1+x)^{-n}$, $n\in\mathbb{N}$, the authors replaced $1/n$ by $\lambda_{n}$, consequently $\lambda=(\lambda_{n})_{n\geq 1}$ satisfies relation (2.1). The new operators are expressed for $x\geq 0$ as follows

 $\displaystyle(V_{n}^{\langle\lambda\rangle}f)(x)=(1+x)^{-1/\lambda_{n}}f(0)$ $\displaystyle+(1+x)^{-1/\lambda_{n}}\sum_{k=1}^{\infty}\displaystyle\frac{1}{k% !}\,\displaystyle\frac{1}{\lambda_{n}}\!\left(\!1+\displaystyle\frac{1}{% \lambda_{n}}\!\right)\!\ldots\!\left(\!k\!-\!1+\displaystyle\frac{1}{\lambda_{% n}}\right)\!\!\left(\displaystyle\frac{x}{1+x}\right)^{k}\!f(k\lambda_{n})$ (2.11)

Regarding these operators, the authors proved [11, Corollary 2] the following relations

 $(V_{n}^{\langle\lambda\rangle}e_{0})(x)=1,{\mathcal{M}}_{1}(V_{n}^{\langle% \lambda\rangle};x)=0,{\mathcal{M}}_{2}(V_{n}^{\langle\lambda\rangle};x)=% \lambda_{n}x(1+x),\ x\geq 0.$

Comparing these relations with the requirements (2.4), (2.5), (2.6), we conclude $l_{1}(x)=l_{3}(x)=0$ and $l_{2}(x)=x(1+x)$, $x\geq 0$.

The space of real valued uniformly continuous functions defined on $\mathbb{R}_{+}$ is denoted by $UC(\mathbb{R}_{+})$.

###### Theorem 2.3.

Let the operators $L_{n}^{\langle\lambda\rangle}$, $n\in\mathbb{N}$, be defined by (2.2) such that the relations (2.4) and (2.6) hold. For $f\in UC(\mathbb{R}_{+})$, $x\geq 0$, we have

 $|(L_{n}^{\langle\lambda\rangle}f)(x)-f(x)|\leq\left(1+\sqrt{l_{2}(x)+\lambda_{% n}l_{3}(x)}\right)\omega\left(f;\sqrt{\lambda_{n}}\right).$ (2.12)

Proof. First of all we mention that $\omega(f;\cdot)$ is well defined for any $f\in UC(\mathbb{R}_{+})$, see, e.g., [8, Chapter 2, §6]. The quantitative result (2.12) given in terms of the modulus of continuity is a direct consequence of the following statement proved by Shisha and Mond [19]: if $\Lambda$ is a linear positive operator defined on $C(K)$, $K$ compact, then one has

 $\displaystyle|(\Lambda f)(x)-f(x)|$ $\displaystyle\leq|f(x)|\,|(\Lambda e_{0})(x)-1|$ $\displaystyle+\left((\Lambda e_{0})(x)+\displaystyle\frac{1}{\delta}\sqrt{(% \Lambda e_{0})(x)(\Lambda\varphi_{x}^{2})(x)}\right)\omega(f;\delta)$ (2.13)

for every $x\in K$ and $\delta>0$.

Even if this result was proved for continuous functions defined on compact intervals, the reasonings are the same for functions belonging to $UC(\mathbb{R}_{+})$. In (2) we choose $\Lambda=L_{n}^{\langle\lambda\rangle}$. Taking into account that $L_{n}^{\langle\lambda\rangle}$ operators reproduce the constants and relation (2.6) holds, we can write

 $|(L_{n}^{\langle\lambda\rangle}f)(x)-f(x)|\leq\left(1+\displaystyle\frac{1}{% \delta}\sqrt{{\mathcal{M}}_{2}(L_{n}^{\langle\lambda\rangle};x)}\right)\omega(% f;\delta),\ x\geq 0.$

Taking $\delta=\sqrt{\lambda_{n}}$, we get inequality (2.12) and the proof is ended. $\hfill\square$

Returning to the two examples, relation (2.12) becomes as follows:

(i) For Szász-type operators (2.9),

 $|T_{n}^{*}(f;\lambda_{n})(x)-f(x)|$
 $\leq\left(1+\sqrt{(H^{\prime\prime}(1)+1)x+\lambda_{n}\displaystyle\frac{A^{% \prime}(1)+A^{\prime\prime}(1)}{A(1)}}\right)\omega\left(f;\sqrt{\lambda_{n}}% \right),$

where $\lambda_{n}=\beta_{n}/\alpha_{n}$.

The result coincides with that obtained in [10, Theorem 2.2].

(ii) For Baskakov type operators (2),

 $|V_{n}^{\langle\lambda\rangle}f)(x)-f(x)|\leq\left(1+\sqrt{x(x+1)}\right)% \omega\left(f;\sqrt{\lambda_{n}}\right).$

The result is similar with that obtained in [11, Corollary 2.1(ii)].

In what follows, we will show that the operators of the form $L_{n}^{\langle\lambda\rangle}$, $n\in\mathbb{N}$, defined by (2.2), may give an arbitrary small order of approximation for quantitative estimates in Voronovskaya type theorems too. Let us recall that Voronovskaya type results for linear positive operators applied to functions defined on $[0,+\infty)$ were obtained, for example, in [12] for operators reproducing $e_{0}$ and $e_{1}$ and in [2] for operators reproducing only $e_{0}$. For the simplicity of presentation, we will use below the results in [12] applied to the above particular cases.

For this purpose, firstly we need the following notations: for $m\in\mathbb{N}$ with $m\geq 2$ and $k\in\mathbb{N}\cup\{0\}$, the Pǎltǎnea [18] modulus of continuity is given by

 $\omega_{\varphi_{m}}(f;h)=\sup\{|f(x)-f(y)|;x,y\geq 0,|x-y|\leq h\cdot\varphi_% {m}\left(\frac{x+y}{2}\right)\},h\geq 0,$

where $\varphi_{m}(x)=\displaystyle\frac{\sqrt{x}}{1+x^{m}}$, for all $x\in[0,+\infty)$,

 $C^{k}[0,+\infty)=\{f:[0,+\infty)\to\mathbb{R};\exists f^{(k)}\mbox{ continuous% on }[0,+\infty)\},$
 $C_{k}[0,+\infty)$
 $=\{f\in C[0,+\infty);\exists M>0,\mbox{ such that }|f(x)|\leq M(1+x^{k}),% \forall x\geq 0\},$
 $W_{m}[0,+\infty)$
 $=\left\{f:[0,+\infty)\to\mathbb{R};f\circ e_{2},f\circ e_{v}\in UC(\mathbb{R}_% {+}),\mbox{ for }v=\frac{1}{2m+1}\right\}.$
###### Theorem 2.4.

Let the Szász-type operators $T^{*}_{n}(f;\lambda_{n})(x)$ be given by (2.9) and for the classical case, $n\in\mathbb{N}$, $m\geq 2$ and $\lambda_{n}\searrow 0$ as fast we want.

If $k=\max\{m+3,6,2m\}$, then $T^{*}_{n}(f;\lambda_{n})(x)$ is well defined on $C_{k}[0,+\infty)$ and for all $f\in C^{2}[0,+\infty)\cap C_{k}[0,+\infty)$, with $f^{\prime\prime}\in W_{m}[0,+\infty)$, all $x\in(0,+\infty)$ and $n\in\mathbb{N}$, we have

 $\left|T^{*}_{n}(f;\lambda)(x)-f(x)-\lambda_{n}\cdot\frac{xf^{\prime\prime}(x)}% {2}\right|$
 $\leq\left(\lambda_{n}x+\sqrt{2A^{*}_{n,m}(x)}\right)\cdot\omega_{\varphi_{m}}% \left(f;\lambda_{n}^{3/2}\cdot\sqrt{\lambda_{n}^{2}+25x\lambda_{n}+15x^{2}}% \right).$ (2.14)

Here $A^{*}_{n,m}(x)=T^{*}_{n}\left(\left[1+\left(x+\frac{|t-x|}{2}\right)^{m}\right% ]^{2};\lambda\right)(x)$, is bounded with respect to $n\in\mathbb{N}$ for any fixed $x\in(0,+\infty)$ and $m\geq 2$, and can be explicitly calculated by a similar formula with that in [12], formula (3.2).

Proof. It is a consequence of Theorem 2.2 in [12], reasoning as in the proof of Theorem 3.1 there, but for the operator $T^{*}_{n}(f;\lambda_{n})(x)$ instead of the classical Szász-type operator. $\hfill\square$

###### Theorem 2.5.

Let the Baskakov-type operators $V_{n}^{\langle\lambda\rangle}f)(x)$ be given by (2), $n\in\mathbb{N}$, $m\geq 2$ and $\lambda_{n}\searrow 0$ as fast we want.

If $k=\max\{m+3,4\}$, then $V_{n}^{\langle\lambda\rangle}f)(x)$ is well defined on $C_{k}[0,+\infty)$ and for all $f\in C^{2}[0,+\infty)\cap C_{k}[0,+\infty)$, with $f^{\prime\prime}\in W_{m}[0,+\infty)$, all $x\in(0,+\infty)$ and $n\in\mathbb{N}$, we have

 $\left|V_{n}^{\langle\lambda\rangle}f)(x)-f(x)-\lambda_{n}\cdot\frac{x(1+x)f^{% \prime\prime}(x)}{2}\right|$
 $\leq\frac{\lambda_{n}}{2}\cdot\left[x(1+x)+\sqrt{2}\sqrt{x}(1+x)C_{n,2,m}(x)\right]$
 $\cdot\omega_{\varphi_{m}}\left(f;\sqrt{\lambda_{n}}\cdot\sqrt{(1+6x+6x^{2})% \lambda_{n}+3x(1+x)}\right).$ (2.15)

Here $C_{n,2,m}(x)$ is bounded with respect to $n\in\mathbb{N}$ for any fixed $x\in(0,+\infty)$ and $m\geq 2$, and can be explicitly calculated by a similar formula with that in the statement of Theorem 2.3 in [12].

Proof. It is a consequence of Theorem 2.3 in [12], reasoning as in the proof of Theorem 3.5 there, but for the operator $V_{n}^{\langle\lambda\rangle}f)(x)$ instead of the classical Baskakov-type operator. $\hfill\square$

The above two theorems are new.

## 3. A Kantorovich type extension

Starting from the operators $L_{n}^{\langle\lambda\rangle}$, $n\in\mathbb{N}$, designed as in (2.2) which verify the assumptions (2.4), (2.5), (2.6), we associate them the following integral generalization of Kantorovich type

 $(K_{n}^{\langle\lambda\rangle}f)(x)=\displaystyle\frac{1}{\lambda_{n}}\sum_{k=% 0}^{\infty}a_{k}(\lambda_{n};x)\int_{k\lambda_{n}}^{(k+1)\lambda_{n}}f(t)dt,\ % x\geq 0.$ (3.1)

The construction involves the mean values of the approximating function $f$ on the subintervals $[k\lambda_{n},(k+1)\lambda_{n}]$, $k\in\mathbb{N}_{0}$.

In the above $f$ must be locally integrable function on $\mathbb{R}_{+}$ such that the antiderivative of $f$ belongs to the domain ${\mathcal{D}}$ defined at (2.3). Our aim is to show that this linear and positive integral process can approximate certain classes of functions with an arbitrary good order of approximation on each compact interval included in $\mathbb{R}_{+}$.

###### Lemma 3.1.

Let the operators $K_{n}^{\langle\lambda\rangle}$, $n\in\mathbb{N}$, be defined by (3.1). The following identities

 $\displaystyle(K_{n}^{\langle\lambda\rangle}e_{0})(x)=1,\quad(K_{n}^{\langle% \lambda\rangle}e_{1})(x)=(L_{n}^{\langle\lambda\rangle}e_{1})(x)+\displaystyle% \frac{1}{2}\lambda_{n},$ $\displaystyle(L_{n}^{\langle\lambda\rangle}e_{2})(x)=(L_{n}^{\langle\lambda% \rangle}e_{2})(x)+\lambda_{n}(L_{n}^{\langle\lambda\rangle}e_{1})(x)+% \displaystyle\frac{1}{3}\lambda_{n}^{2}$

hold for each $x\geq 0$.

Proof. The first identity is a consequence of relation (2.4). Further on, we can write

 $\displaystyle(K_{n}^{\langle\lambda\rangle}e_{1})(x)$ $\displaystyle=\displaystyle\frac{1}{\lambda_{n}}\sum_{k=0}^{\infty}a_{k}(% \lambda_{n};x)\left(k+\displaystyle\frac{1}{2}\right)\lambda_{n}^{2}=(L_{n}^{% \langle\lambda\rangle}e_{1})(x)+\displaystyle\frac{1}{2}\lambda_{n}$

and

 $\displaystyle(K_{n}^{\langle\lambda\rangle}e_{2})(x)$ $\displaystyle=\displaystyle\frac{1}{\lambda_{n}}\sum_{k=0}^{\infty}a_{k}(% \lambda_{n};x)\left(k^{2}+k+\displaystyle\frac{1}{3}\right)\lambda_{n}^{3}$ $\displaystyle=(L_{n}^{\langle\lambda\rangle}e_{2})(x)+\lambda_{n}(L_{n}^{% \langle\lambda\rangle}e_{1})(x)+\displaystyle\frac{1}{3}\lambda_{n}^{2}.$

$\hfill\square$

By using Lemma 3.1 we can express the first and the second central moments of $K_{n}^{\langle\lambda\rangle}$ operators with the help of corresponding moments of $L_{n}^{\langle\lambda\rangle}$. For each $x\geq 0$ we get

 ${\mathcal{M}}_{1}(K_{n}^{\langle\lambda\rangle};x)={\mathcal{M}}_{1}(L_{n}^{% \langle\lambda\rangle};x)+\displaystyle\frac{1}{2}\lambda_{n},$ (3.2)
 ${\mathcal{M}}_{2}(K_{n}^{\langle\lambda\rangle};x)={\mathcal{M}}_{2}(L_{n}^{% \langle\lambda\rangle};x)+\lambda_{n}{\mathcal{M}}_{1}(L_{n}^{\langle\lambda% \rangle};x)+\displaystyle\frac{1}{3}\lambda_{n}^{2}.$ (3.3)

A question arises: are well defined the operators $L_{n}^{\langle\lambda\rangle}$ and $K_{n}^{\langle\lambda\rangle}$ for functions of $UC(\mathbb{R}_{+})$? Since $f$ belongs to this class, its growth is linear, i.e., the non-negative real constants $c_{1}$ and $c_{2}$ exist such that $|f(x)|\leq c_{1}x+c_{2}$, $x\geq 0$. Consequently, the answer is positive.

###### Theorem 3.2.

Let the operators $K_{n}^{\langle\lambda\rangle}$, $n\in\mathbb{N}$, be defined by (3.1) such that the relations (2.4), (2.5) and (2.6) hold. For every $f\in UC(\mathbb{R}_{+})$

 $|(K_{n}^{\langle\lambda\rangle}f)(x)-f(x)|\!\leq\!\left(\!1\!+\!\sqrt{l_{2}(x)% +\left(l_{1}(x)+l_{3}(x)+\displaystyle\frac{1}{3}\right)\lambda_{n}}\right)% \omega\left(f;\sqrt{\lambda_{n}}\right)$ (3.4)

takes place, where $l_{i}$, $1\leq i\leq 3$, are given at relations (2.4), (2.5), (2.6).

Proof. We apply the reasoning used at the proof of Theorem 2.3. Taking into account that $K_{n}^{\langle\lambda\rangle}e_{0}=e_{0}$ and relations (3.2), (3.3), inequality (2) becomes as follows

 $|(K_{n}^{\langle\lambda\rangle}f)(x)-f(x)|\!\leq\!\!\left(\!\!1\!+% \displaystyle\frac{1}{\delta}\sqrt{{\mathcal{M}}_{2}(L_{n}^{\langle\lambda% \rangle};x)\!+\!\lambda_{n}{\mathcal{M}}_{1}(L_{n}^{\langle\lambda\rangle};x)% \!+\!\displaystyle\frac{1}{3}\lambda_{n}^{2}}\right)\!\omega(f;\delta).$

Further, we use the identities (2.4), (2.5), (2.6) and choose $\delta:=\sqrt{\lambda_{n}}$. After a few calculations we arrive at the desired relation. $\hfill\square$

It is known that the function $\omega$ defined by (1.2) is continuous at $\delta=0$, i.e.

 $\lim\limits_{\delta\to 0^{+}}\omega(f;\delta)=\omega(f;0)=0,$

if and only if $f$ is uniformly continuous on its domain, see [8, page 40]. Examining relations (3.4) and (2.1), the following pointwise convergence

 $\lim\limits_{n\to\infty}(K_{n}^{\langle\lambda\rangle}f)(x)=f(x),\ x\in\mathbb% {R}_{+},$

takes place. These operators can also uniformly approximate any function $f\in UC(\mathbb{R}_{+})$ on each compact, say $K$, included in $\mathbb{R}_{+}$. Relative to the functions $l_{i}\in C(\mathbb{R}_{+})$, $i\in\{1,2,3\}$, introduced by relations (2.4)-(2.6), we denote

 $\max_{x\in K}l_{2}(x)=\alpha,\quad\max_{x\in K}\left(l_{1}(x)+l_{3}(x)+% \displaystyle\frac{1}{3}\right)=\beta.$

Relation (3.4) implies

 $|(K_{n}^{\langle\lambda\rangle}f)(x)-f(x)|\leq\left(1+\sqrt{\alpha+\beta% \lambda_{n}}\right)\omega\left(f;\sqrt{\lambda_{n}}\right),\ x\in K,$

therefore $\lim\limits_{n\to\infty}K_{n}^{\langle\lambda\rangle}f(x)=f(x)$, uniformly on the compact $K$.

We recall $f:\mathbb{R}_{+}\to\mathbb{R}$ is Lipschitz continuous if there exists a real constant $M\geq 0$ such that $|f(x)-f(y)|\leq M|x-y|$ for any $(x,y)\in\mathbb{R}_{+}\times\mathbb{R}_{+}$. We denote this set by $Lip_{M}1$. One hand, any Lipschitz continuous map is uniformly continuous; on the other hand, $f\in Lip_{M}1$ if and only if $\omega_{f}(\delta)\leq M\delta$. Taking in view these statements, we enunciate

###### Corollary 3.3.

Let the operators $K_{n}^{\langle\lambda\rangle}$, $n\in\mathbb{N}$, be defined by (3.1) such that the relations (2.4), (2.5) and (2.6) hold. For every $f\in Lip_{M}1$, $M\geq 0$ and $x\geq 0$, it takes place

 $|(K_{n}^{\langle\lambda\rangle}f)(x)-f(x)|\leq M\left(1+\sqrt{l_{2}(x)+\left(l% _{1}(x)+l_{3}(x)+\displaystyle\frac{1}{3}\right)\lambda_{n}}\right)\sqrt{% \lambda_{n}}.$
###### Example 3.4.

Let us consider the Baskakov-Kantorovich operators, defined by the formula

 $BK_{n}^{\langle\lambda\rangle}(f)(x)$
 $=\!(1+x)^{-\frac{1}{\lambda_{n}}}\sum_{j=0}^{\infty}\frac{1}{j!}\frac{1}{% \lambda_{n}}\!\left(\!\!1\!+\!\frac{1}{\lambda_{n}}\right)\!\ldots\!\left(\!j% \!-\!1+\frac{1}{\lambda_{n}}\right)\!\frac{x^{j}}{(1+x)^{j}}\frac{1}{\lambda_{% n}}\int_{j\lambda_{n}}^{(j+1)\lambda_{n}}\!f(v)dv.$

Since $BK_{n}^{\langle\lambda\rangle}((t-x)^{2})(x)=\lambda_{n}\left(x^{2}+x+\frac{1}% {3}\cdot\lambda_{n}\right)$, by Theorem 3.2 and Corollary 3.3 we get the estimates

 $|BK_{n}^{\langle\lambda\rangle}(f)(x)-f(x)|\leq 2\omega_{1}\left(f;\sqrt{% \lambda_{n}}\cdot\sqrt{x^{2}+x+\lambda_{n}/3}\right)$

and

 $|BK_{n}^{\langle\lambda\rangle}(f;\lambda_{n})(x)-f(x)|\leq 2M\sqrt{\lambda_{n% }}\cdot\sqrt{x^{2}+x+\lambda_{n}/3},$

respectively, which are similar to Theorem 2.2 and Corollary 2.3 in Trifa [21].

###### Remark 3.5.

Analyzing the proofs of Theorems 2.3 and 3.2, we deduce that the upper bound error of approximation could be established if we relax requirements (2.5) and (2.6). More precisely, we can replace them with the followings

 $0\leq{\mathcal{M}}_{1}(L_{n}^{\langle\lambda\rangle};x)\leq\lambda_{n}l_{1}(x),$ (3.5)
 ${\mathcal{M}}_{2}(L_{n}^{\langle\lambda\rangle};x)\leq\lambda_{n}l_{2}(x)+% \lambda_{n}^{2}l_{3}(x),$ (3.6)

where $l_{j}\in C(\mathbb{R}_{+})$, $j\in\{1,2,3\}$.

In what follows, we present a class of operators which has not been yet approached from the perspective of our work. Also we mention that this class of operators restarted to be studied in various recent papers, see, i.e., [1], [7], [9], published in 2016.

###### Example 3.6.

The starting point is the following Poisson-type distribution

 $\omega_{\beta}(k,\alpha)=\displaystyle\frac{\alpha}{k!}(\alpha+k\beta)^{k-1}e^% {-(\alpha+k\beta)},\ k\in\mathbb{N}_{0},$

where $\alpha>0$ and $|\beta|<1$. Jain [14] introduced the sequence of linear operators by formula

 $(P_{n}^{[\beta]}f)(x)=\sum_{k=0}^{\infty}\omega_{\beta}(k,nx)f\left(% \displaystyle\frac{k}{n}\right),\ f\in C(\mathbb{R}_{+}),$

where $0\leq\beta<1$. It was been shown [14, Lemma 1]

 $\sum_{k=0}^{\infty}\omega_{\beta}(k,\alpha)=1.$ (3.7)

In order to become $(P_{n}^{[\beta]})_{n\geq 1}$ an approximation process, $\beta$ should depend on $n$ and its limit to be zero. We modify these operators following the model described by (2.2). Consequently, we choose $\beta=\lambda_{n}$, $n\geq 1$, a strictly decreasing sequence of positive numbers with the property (2.1). By following the notation used in this paper, we consider the modified operators

 $(P_{n}^{\langle\lambda\rangle}f)(x)=\sum_{k=0}^{\infty}\omega_{\lambda_{n}}% \left(k,\displaystyle\frac{x}{\lambda_{n}}\right)f(k\lambda_{n}).$

Based on (3.7) and [14, Eqs. (2.13), (2.14)], we get

 $\displaystyle(P_{n}^{\langle\lambda\rangle}e_{0})(x)=1,$ $\displaystyle{\mathcal{M}}_{1}(P_{n}^{\langle\lambda\rangle};x)=\displaystyle% \frac{\lambda_{n}}{1-\lambda_{n}}x,$ $\displaystyle{\mathcal{M}}_{2}(P_{n}^{\langle\lambda\rangle};x)=\displaystyle% \frac{\lambda_{n}^{2}}{(1-\lambda_{n})^{2}}x^{2}+\displaystyle\frac{\lambda_{n% }}{(1-\lambda_{0})^{3}}x.$

Since $(\lambda_{n})_{n\geq 1}$ is a positive strictly decreasing sequence, we can write

 $\displaystyle 0\leq\$ $\displaystyle{\mathcal{M}}_{1}(P_{n}^{\langle\lambda\rangle};x)\leq% \displaystyle\frac{x}{1-\lambda_{1}}\lambda_{n},$ $\displaystyle{\mathcal{M}}_{2}(P_{n}^{\langle\lambda\rangle};x)\leq% \displaystyle\frac{x}{(1-\lambda_{1})^{3}}\lambda_{n}+\displaystyle\frac{x^{2}% }{(1-\lambda_{1})^{2}}\lambda_{n}^{2}.$

Considering formula (3.1), Kantorovich type extension will be read as follows

 $(\widetilde{K}_{n}^{\langle\lambda\rangle}f)(x)=\displaystyle\frac{1}{\lambda_% {n}}\sum_{k=0}^{\infty}\omega_{\lambda_{n}}\left(k,\displaystyle\frac{x}{% \lambda_{n}}\right)\int_{k\lambda_{n}}^{(k+1)\lambda_{n}}f(t)dt,\ x\geq 0.$

Set $\tau=(1-\lambda_{1})^{-1}$. For every $f\in UC(\mathbb{R}_{+})$, in the light of relations (3.5) and (3.6), Theorem 2.3 implies

 $|(\widetilde{P}_{n}^{\langle\lambda\rangle}f)(x)-f(x)|\leq\left(1+\tau\sqrt{x(% \tau+\lambda_{n}x)}\right)\omega\left(f;\sqrt{\lambda_{n}}\right)$

and Theorem 3.2 implies

 $|(\widetilde{K}_{n}^{\langle\lambda\rangle}f)(x)-f(x)|\leq\left(1+\sqrt{\tau^{% 3}x+\left(\tau^{2}x^{2}+\tau x+\frac{1}{3}\right)\lambda_{n}}\right)\omega% \left(f;\sqrt{\lambda_{n}}\right).$
###### Remark 3.7.

For the Kantorovich type operators in this section, we can use the results in [2] in order to obtain for them quantitative estimates of arbitrarily small rate.

###### Remark 3.8.

Usually, the linear positive operators provide an approximation error of order ${\mathcal{O}}\left(\sqrt{1/n}\right)$. This rate of convergence can be modified by the mean of formulas (2.2), (3.1) such that its magnitude to be of order ${\mathcal{O}}\left(\sqrt{\lambda_{n}}\right)$, where $(\lambda_{n})_{n\geq 1}$ can be chosen to converge arbitrarily fast to zero. Also, the results obtained have a strong unifying character, in the sense that for various choices of the nodes $\lambda_{n}$, one can recapture previous approximation results obtained for these operators by other authors, see, e.g., [6], [22], [23]. Finally, we mention that our results here target only uniformly continuous functions defined on unbounded intervals.

###### Remark 3.9.

The method in this paper could be applied in approximation by radial basis functions too, as follows. In the paper [16], it was formally introduced the sequences of the multivariate operators of the form

 $L_{n}(f)(x)=\frac{1}{n^{s(1-r)}}\sum_{j\in I_{n}(\Omega_{h})}f(j/n)\Phi(nx-jn^% {r-1}),$

where $x\in\mathbb{R}^{s}$, $j$ is a multiindex and

 $I_{n}(\Omega_{h})=\{j\in\mathbb{Z}^{s};[j/n,(j+1)/n]^{s}\cap\Omega_{h}\not=% \emptyset\}.$

Under the hypothesis in Theorem 2.2 in [16], the following quantitative estimate was obtained

 $\|L_{n}(f)-f\|_{L^{p}(\Omega)}\leq\frac{c}{n^{\gamma}}\cdot\|f\|_{W^{p,1}(% \Omega_{h})},$

where $\Omega$ and $\Omega_{h}$ are bounded domains.

Now, choosing $\lambda_{n}\searrow 0$ as fast we want and defining

 $L_{n}(f;\lambda_{n})(x)=\lambda_{n}^{s(1-r)}\sum_{j\in I_{\lambda_{n}}(\Omega_% {h})}f(j\lambda_{n})\Phi(x/\lambda_{n}-j/\lambda_{n}^{r-1}),$

reasoning exactly as in the proof of Theorem 2.2 in [16] and under the same hypothesis, one gets the much better estimate

 $\|L_{n}(f;\lambda_{n})-f\|_{L^{p}(\Omega)}\leq c\lambda_{n}^{\gamma}\cdot\|f\|% _{W^{p,1}(\Omega_{h})}.$

## References

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1. Abel, U., Agratini, O.: Asymptotic behaviour of Jain operators. Numer. Algorithm 71(3), 553–565 (2016)
2. Aral, A., Gonska, H., Heilmann, M., Tachev, G.: Quantitative Voronovskaya-type results for polynomially bounded functions. Results Math. 70, 313–324 (2016)
3. Baskakov, V.A.: An example of a sequence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk SSSR 113, 249–251 (1957). (in Russian)
4. Bleimann, G., Butzer, P.L., Hahn, L.A.: Bernstein-type operators approximating continuous functions on the semiaxis. Indag. Math. 42, 255–262 (1980)
5. Bohman, H.: On approximation of continuous and of analytic functions. Ark. Mat. 2(1952–1054), 43–56
6. Cetin, N., Ispir, N.: Approximation by complex modified Szász-Mirakjan operators. Studia Sci. Math. Hungar. 50(3), 355–372 (2013)
7. Deniz, E.: Quantitative estimates for Jain-Kantorovich operators. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 65(2), 121–132 (2016)
8. DeVore, R.A., Lorentz, G.G.: Constructive Approximation, Grundlehren der Mathematischen Wissenschaften 303. Springer, Berlin (1993)
9. Dogru, O., MohapatraR. N., Örkü, M.:Approximation properties of generalized Jain operators. Filomat 30(1016), 2359-2366
10. Gal, S.G.: Approximation with an arbitrary order by generalized Szász-Mirakjan operators. Studia Univ. Babes-Bolyai Math. 59(1), 77–81 (2014)
11. Gal, S.G., Opris, B.D.: Approximation with an arbitrary order by modified Baskakov type operators. Appl. Math. Comput. 265, 329–332 (2015)
12. Gupta, V., Tachev, G.: General form of Voronovskaja’s theorem in terms of weighted modulus of continuity. Results Math. 69, 419–430 (2016)
13. Ismail, M.E.H.: On a generalization of Szász operators. Mathematica (Cluj) 39, 259–267 (1974)
14. Jain, G.C.: Approximation of functions by a new class of linear operators. J. Aust. Math. Soc. 13(3), 271–276 (1972)
15. Korovkin, P.P.: On convergence of linear positive operators in space of continuous function. Dokl. Akad. Nauk SSSR (N.S.) 90, 961–964 (1953). (in Russian)
16. Mastroianni, G.: Su un operatore lineare e positivo. Rend. Acc. Sc. Fis. Mat. Napoli Serie IV 46, 161–176 (1979)
17. Paltanea, R.: Estimates of approximation in terms of a weighted modulus of continuity. Bull. Transilvania Univ. Brasov 4(53), 67–74 (2011)
18. Shisha, O., Mond, B.: The degree of convergence of linear positive operators. Proc. Nat. Acad. Sci. USA 60, 1196–1200 (1968)
19. Szász, O.: Generalization of S.Bernstein polynomials to the infinite interval. J. Res. Nat. Bur. Standards 45, 239–245 (1950)
20. Trifa, S.: Approximation with an arbitrary order by generalized Kantorovich-type and Durrmeyer-type operators on [0+∞). Studia Univ. Babes-Bolyai Math. (2017) 62(2) (in print)
21. Walczak, Z.: On approximation by modified Szász-Mirakjan operators. Glas. Mat. 37(57), 303–319 (2002)
22. Walczak, Z.: On modified Szász-Mirakjan operators. Novi Sad J. Math. 33(1), 93–107 (2003)