## Abstract

This note aims to highlight a general class of discrete linear and positive operators, focusing on some of their approximation properties. The construction includes a sequence of positive numbers ðknÞ, strictly decreasing. Imposing additional conditions on it, we set the convergence of the operators towards the identity operator and establish upper bounds for the error of approximation. A probabilistic approach is given. Also, in certain weighted spaces the study of its approximation properties is developed.

## Authors

**Octavian Agratini**

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

## Keywords

Abdulofizov generalization; Balazs-Szabados operator; Korovkin theorem; modulus of smoothness.

## Paper coordinates

O. Agratini, *On a class of Bernstein-type rational functions*, Numerical Functional Analysis and Optimization, 41 (2020) no. 4, pp. 483-494, https://doi.org/10.1080/01630563.2019.1664566

requires subscription: https://doi.org/10.1080/01630563.2019.1664566

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# On a class of Bernstein type rational functions

###### Abstract.

This note aims to highlight a general class of discrete linear and positive operators, focusing on some of their approximation properties. The construction includes a sequence of positive numbers $({\lambda}_{n})$, strictly decreasing. Imposing additional conditions on it, we set the convergence of the operators towards the identity operator and establish upper bounds for the error of approximation. A probabilistic approach is given. Also, in certain weighted spaces the study of its approximation properties is developed.

Keywords and phrases: Linear positive operator, Korovkin theorem, modulus of smoothness, Balázs-Szabados operator, Abdulofizov generalization.

Mathematics Subject Classification: Primary: 41A25, Secondary: 47A63.

## 1. Introduction

The classical Korovkin theory is mainly connected with the approximation of continuous functions by linear operators having as essential ingredient the property of positivity. Considering such operators, say ${L}_{n}$ $(n\in \mathbb{N})$, they have the approximation property on the normed space $(S,\parallel \cdot {\parallel}_{S})$ if $\underset{n\to \mathrm{\infty}}{lim}{\Vert {L}_{n}f-f\Vert}_{S}=0$ for each $f\in S$.

In what follows, we consider linear positive operators of discrete type which approximate functions defined on ${\mathbb{R}}_{+}=[0,\mathrm{\infty})$. In order to construct the proposed class of operators, we use a strictly decreasing positive sequence ${({\lambda}_{n})}_{n\ge 1}$ with the property

$$\underset{n\to \mathrm{\infty}}{lim}{\lambda}_{n}=0.$$ | (1) |

The operators investigated in this paper are designed as follows

$$({L}_{n}f)(x)=\frac{1}{{(1+{\lambda}_{n}x)}^{n}}\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right){({\lambda}_{n}x)}^{k}f\left(\frac{k}{n{\lambda}_{n}}\right),n\in \mathbb{N},x\ge 0,$$ | (2) |

where $f$ is continuous function on ${\mathbb{R}}_{+}$ satisfying a certain growth condition. We also require

$$\underset{n\to \mathrm{\infty}}{lim}n{\lambda}_{n}=\mathrm{\infty}$$ | (3) |

to be fulfilled.

The construction given at (2) is not new. With a slight modification, it appears in [7, Eq. (1.2)]. In this mentioned paper, K. Balázs studied the special case ${\lambda}_{n}={n}^{-1/3}$. In time, another special case has been studied deeper

$$ | (4) |

and for these particular operators we use the notation ${R}_{n}^{[\beta ]}$. We point out some of the most outstanding results concerning these operators. Balázs and Szabados [8] gave weighted estimated and investigated the uniform convergence of ${R}_{n}^{[\beta ]}$. V. Totik [19] settled the saturation properties of ${R}_{n}^{[\beta ]}$, proved a general convergence theorem for ${R}_{n}^{[\beta ]}$ – like rational functions and obtained a Voronovskaja-type result. Biancamaria Della Vecchia [9] proved some preservation properties and asymptotic relations for ${R}_{n}^{[\beta ]}$. Further on, Ulrich Abel and B. Della Vecchia [4] obtained the complete asymptotic expansion for ${R}_{n}^{[\beta ]}$, all coefficients being calculated explicitly. An integral generalization in Kantorovich sense was achieved in [5].

In Quantum Calculus, the $q$-analogues of ${R}_{n}^{[\beta ]}$ operators were studied by O. Doğru [11]. The rational complex ${R}_{n}^{[\beta ]}$ operators, $$, appear in the monograph of S.G. Gal [13]. See, respectively, the papers of Ispir and Yildiz Ozkan [14], Mahmudov [17].

Returning at general form indicated by relation (2), in the recent paper [3] the authors showed that ${L}_{n}$, $n\in \mathbb{N}$, operators enjoy the variation detracting property. At this moment, we want to point out some features of ${L}_{n}$, $n\in \mathbb{N}$, operators. They are built using a finite sum. Other classes of discrete operators are expressed by series, making them harder to use in approximation of functions, see for example, Baskakov operators or Favard-Szász-Mirakjan operators. For each $n\in \mathbb{N}$, ${L}_{n}$ uses an equidistant net ${\mathrm{\Delta}}_{n}={(k/(n{\lambda}_{n}))}_{k=\overline{0,n}}$ with a flexible step, that is ${(n{\lambda}_{n})}^{-1}$. It changes depending on an arbitrary sequence ${({\lambda}_{n})}_{n\ge 1}$ satisfying the conditions (1) and (3).

Regarding the definition of ${L}_{n}$, $n\in \mathbb{N}$, operators, it is fair to mention that a more general form was proposed a long time ago by Abdulofizov [1]. The author considered the linear operators ${S}_{n}$, $n\in \mathbb{N}$, defined as follows

$$({S}_{n}f)(x)=\sum _{k=0}^{n}{q}_{k,n}(x)f({\xi}_{k,n}),$$ |

where

$${q}_{k,n}(x)=\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{{(1+{r}_{n})}^{k}{x}^{k}}{{({r}_{n}(1+x))}^{n}}{({r}_{n}-x)}^{n-k}\text{and}{\xi}_{k,n}=\frac{k}{n-k+n{r}_{n}^{-1}},k=\overline{0,n}.$$ |

Instead of the net used in (2), here appears a net of more general nodes. Estimations of $|({S}_{n}f)(x)-f(x)|$ under various assumptions concerning $f$ have been proved by Abdulofizov. Modifications of Bernstein operators by using nodes of the form ${({\xi}_{k,n})}_{1\le k\le n}$ can also be found in [2].

The approximation properties highlighted in our paper are different from those obtained in [1]. More precisely, we establish two variants of local upper bounds of the error of approximation, one of them including the distance $d(x,E)$ between a certain $x>0$ and a subset $E\subset {\mathbb{R}}_{+}$. Also, we prove that our class of operators does not always form an approximation process in some weighted spaces. Further on, by using a weighted modulus of smoothness defined in 2004, we give both a local and global estimation of the rate of convergence. It is known that a more general class of sequences enjoys fewer specific properties than special cases of that. Particular cases narrow the approach but sometimes provide better approximation variants.

## 2. Preliminary results

Set ${\mathbb{N}}_{0}=\{0\}\cup \mathbb{N}$. For every $m\in {\mathbb{N}}_{0}$ we shall denote by ${e}_{m}$ the monomial defined as ${e}_{0}(t)=1$ and ${e}_{m}(t)={t}^{m}$ $(m\ge 1)$, where $t\in {\mathbb{R}}_{+}$.

Lemma 1. Let ${L}_{n}$, $n\in \mathbb{N}$, be defined by (2). For each $x\in {\mathbb{R}}_{+}$, the following relations

$$({L}_{n}{e}_{0})(x)=1,$$ | (5) |

$$({L}_{n}{e}_{1})(x)=\frac{x}{1+{\lambda}_{n}x},$$ | (6) |

$$({L}_{n}{e}_{2})(x)=\frac{1}{{(1+{\lambda}_{n}x)}^{2}}\left({x}^{2}+\frac{x}{n{\lambda}_{n}}\right)$$ | (7) |

take place.

Proof. Since $\sum _{k=0}^{n}}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){({\lambda}_{n}x)}^{k}={(1+{\lambda}_{n}x)}^{n$, the first identity is evident. Further, we get

$({L}_{n}{e}_{1})(x)$ | $={\displaystyle \frac{1}{{(1+{\lambda}_{n}x)}^{n}}}{\displaystyle \sum _{k=1}^{n}}{\displaystyle \frac{k}{n{\lambda}_{n}}}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){({\lambda}_{n}x)}^{k}$ | ||

$={\displaystyle \frac{x}{{(1+{\lambda}_{n}x)}^{n}}}{\displaystyle \sum _{k=1}^{n}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{k-1}}\right){({\lambda}_{n}x)}^{k-1}$ | |||

$={\displaystyle \frac{x}{1+{\lambda}_{n}x}}.$ |

The last identity is obtained as follows

$({L}_{n}{e}_{2})(x)$ | $={\displaystyle \frac{1}{{(1+{\lambda}_{n}x)}^{n}}}{\displaystyle \sum _{k=1}^{n}}{\displaystyle \frac{{k}^{2}}{{n}^{2}{\lambda}_{n}^{2}}}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right){({\lambda}_{n}x)}^{k}$ | ||

$={\displaystyle \frac{1}{{(1+{\lambda}_{n}x)}^{n}}}\left({\displaystyle \sum _{k=2}^{n}}{\displaystyle \frac{n-1}{n{\lambda}_{n}^{2}}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-2}{k-2}}\right){({\lambda}_{n}x)}^{k}+{\displaystyle \frac{1}{n{\lambda}_{n}^{2}}}{\displaystyle \sum _{k=1}^{n}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{k-1}}\right){({\lambda}_{n}x)}^{k}\right)$ | |||

$={\displaystyle \frac{n-1}{n}}{\displaystyle \frac{{x}^{2}}{{(1+{\lambda}_{n}x)}^{2}}}+{\displaystyle \frac{x}{n{\lambda}_{n}(1+{\lambda}_{n}x)}},$ |

which leads us to the identity (7). $\mathrm{\square}$

Remarks. (i) Since ${L}_{n}$, $n\in \mathbb{N}$, are linear operators, relation (5) implies that they reproduce the constants, consequently ${L}_{n}$ are of Markov type.

(ii) $({L}_{n}f)(0)=f(0)$, accordingly the operators interpolate functions in $x=0$.

We introduce the $s$-th order central moment of the operator ${L}_{n}$, $s\in {\mathbb{N}}_{0}$, that is ${\mu}_{n,s}(x)=({L}_{n}{\phi}_{x,s})(x)$, where

$${\phi}_{x,s}(t)={(t-x)}^{s},x\ge 0,t\ge 0.$$ | (8) |

In view of the identities (5)-(7), by a simple computation, for any $x\ge 0$, we get

$$\{\begin{array}{ccc}{\mu}_{n,0}(x)=1,{\mu}_{n,1}(x)=-\frac{{\lambda}_{n}x}{1+{\lambda}_{n}x},\hfill & & \\ {\mu}_{n,2}(x)=\frac{{\lambda}_{n}^{2}{x}^{4}}{{(1+{\lambda}_{n}x)}^{2}}+\frac{x}{n{\lambda}_{n}{(1+{\lambda}_{n}x)}^{2}}.\hfill & & \end{array}$$ | (9) |

We mention that in the particular case (4), for $s\in \mathbb{N}$ and for all integers $n>{x}^{1/(1-\beta )}$, the moments ${R}_{n}^{[\beta ]}{e}_{s}$ and the central moments ${R}_{n}^{[\beta ]}({(\cdot -x)}^{s},\cdot )$ have been determined in the paper [4, Lemma 3.1, Lemma 3.3].

In the sequel we denote by ${C}_{B}({\mathbb{R}}_{+})$ the Banach lattice of all real-valued bounded and continuous functions on ${\mathbb{R}}_{+}$ endowed with the natural order and the sup-norm $\parallel \cdot {\parallel}_{\mathrm{\infty}}$,

$${\Vert f\Vert}_{\mathrm{\infty}}=\underset{x\ge 0}{sup}|f(x)|.$$ |

If $K$ is a compact interval (in our case $K\subset {\mathbb{R}}_{+}$), the same norm is valid in the space $C(K)$.

We also recall the notion of modulus of smoothness associated with any bounded function $h$ defined on ${\mathbb{R}}_{+}$. For any $\delta \ge 0$, it is given as follows

${\omega}_{h}(\delta )\equiv \omega (h;\delta )$ | $=sup\{|h({x}^{\prime})-h({x}^{\prime \prime})|:{x}^{\prime},{x}^{\prime \prime}\in {\mathbb{R}}_{+},|{x}^{\prime}-{x}^{\prime \prime}|\le \delta \}$ | |||

$=\underset{0\le h\le \delta}{sup}\underset{x\ge 0}{sup}|f(x+h)-f(x)|.$ | (10) |

If $h\in C({\mathbb{R}}_{+})$ and supremum in (2) is taken only for $({x}^{\prime},{x}^{\prime \prime})\in K\times K$, $K\subset {\mathbb{R}}_{+}$ compact interval, we can use the writing option ${\omega}_{K}(h;\cdot )$.

The significant properties of $\omega (h;\cdot )$ are presented, e.g., in [16, pp. 43-46]. Among these, we emphasize that ${\omega}_{h}$ is a non-decreasing function and

$$\omega (h;\mu \delta )\le (1+\mu )\omega (h;\delta ),(\delta ,\mu )\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}.$$ | (11) |

Modulus of smoothness is continuous at $\delta =0$, i.e.

$$\underset{\delta \to {0}^{+}}{lim}\omega (h;\delta )=\omega (h;0)=0,$$ | (12) |

if and only if $h$ is uniformly continuous on its domain, [10, page 40] can be consulted.

At the end of this section we present a probabilistic approach of the studied sequence. To accomplish this, we use a classical scheme that can be found, for example, in [6, Section 5.2]. We are fixing a probability space $(\mathrm{\Omega},\mathcal{F},P)$ and a random scheme on ${\mathbb{R}}_{+}$, $Z:\mathbb{N}\times {\mathbb{R}}_{+}\to {\mathcal{M}}_{2}(\mathrm{\Omega})$, where ${\mathcal{M}}_{2}(\mathrm{\Omega})$ is the space of all real square-integrable random variables on $\mathrm{\Omega}$. ${Z}{(}{n}{,}{x}{)}$ is discretely distributed taking the values ${k}{/}{(}{n}{{\lambda}}_{{n}}{)}$, ${0}{\le}{k}{\le}{n}$, such that

$${P}{\left\{}{Z}{(}{n}{,}{x}{)}{=}\frac{{k}}{{n}{{\lambda}}_{{n}}}{\right\}}{=}{{p}}_{{n}{,}{k}}{(}{x}{)}{,}{\text{where}}{{p}}_{{n}{,}{k}}{(}{x}{)}{=}{\left(}\genfrac{}{}{0pt}{}{{n}}{{k}}{\right)}\frac{{{(}{{\lambda}}_{{n}}{x}{)}}^{{k}}}{{{(}{1}{+}{{\lambda}}_{{n}}{x}{)}}^{{n}}}{.}$$ |

Consequently, for $f\in {C}_{B}({\mathbb{R}}_{+})$ we get

$$({L}_{n}f)(x)=E(f\circ Z(n,x))={\int}_{\mathrm{\Omega}}f\circ Z(n,x)\mathit{d}P={\int}_{\mathbb{R}}f{P}_{Z(n,x)},$$ |

which leads us to relation (2). In the above $E$ is the expected value operator and ${P}_{Z(n,x)}$ represents the distribution of $Z(n,x)$ with respect to $P$. The above identities imply $\Vert {L}_{n}f\Vert \le \Vert f\Vert $.

## 3. Results

We show that the sequence ${({L}_{n})}_{n\ge 1}$ is an approximation process in a certain space, more precisely it takes place

Theorem 1. Let the operators ${L}_{n}$, $n\in \mathbb{N}$, be defined by (2) such that the requirements (1) and (3) are fulfilled. For any compact interval $K\subset {\mathbb{R}}_{+}$, the following relation

$$\underset{n\to \mathrm{\infty}}{lim}{L}_{n}f=f\text{uniformly on}K$$ | (13) |

occurs, provided $f\in C({\mathbb{R}}_{+})$.

Proof. Based on the properties (1) and (3), relations (5)-(7) involve

$$\underset{n\to \mathrm{\infty}}{lim}({L}_{n}{e}_{j})(x)={e}_{j}(x),x\in {\mathbb{R}}_{+},j\in \{0,1,2\}.$$ | (14) |

Further, we defined the lattice homomorphism ${T}_{K}:C({\mathbb{R}}_{+})\to C(K)$ given by ${T}_{K}(f)={f|}_{K}$ for every $f\in C({\mathbb{R}}_{+})$. Identities (14) can be rewritten in following form

$$\underset{n\to \mathrm{\infty}}{lim}{T}_{K}({L}_{n}{e}_{j})={T}_{K}({e}_{j}),j\in \{0,1,2\},$$ |

uniformly on $K$. The Bohman-Korovkin type criterion, see, e.g., [6, Theorem 4.1.4 (vi)], implies our statement (13). $\mathrm{\square}$

The rate of convergence will be established by using modulus of smoothness defined by (2).

Theorem 2. Let the operators ${L}_{n}$, $n\in \mathbb{N}$, be defined by (2) such that the requirements (1) and (3) are fulfilled. For any function $f\in {C}_{B}({\mathbb{R}}_{+})$, we have

$$|({L}_{n}f)(x)-f(x)|\le \left(1+\sqrt{{x}^{4}+x}\right)\omega (f;\sqrt{{\lambda}_{n}^{\ast}}),x\ge 0,$$ | (15) |

where

$${\lambda}_{n}^{\ast}=\mathrm{max}\{{\lambda}_{n}^{2},{(n{\lambda}_{n})}^{-1}\}.$$ | (16) |

Proof. In order to obtain easily this quantitative result we use the following inequality proved by Shisha and Mond [18], that says: if $\mathrm{\Lambda}$ is a linear positive operator defined on $C({\mathbb{R}}_{+})$, then one has

$$|(\mathrm{\Lambda}f)(x)-f(x)|\le |f(x)||(\mathrm{\Lambda}{e}_{0})(x)-1|$$ |

$$+\left((\mathrm{\Lambda}{e}_{0})(x)+\frac{1}{\delta}\sqrt{(\mathrm{\Lambda}{e}_{0})(x)(\mathrm{\Lambda}{\phi}_{x,2})(x)}\right)\omega (f;\delta ),$$ | (17) |

for every $f\in {C}_{B}({\mathbb{R}}_{+})$, $x\in {\mathbb{R}}_{+}$ and $\delta >0$. The function ${\phi}_{x,2}$ was defined by (8). To achieve the condition $$ for every $\delta >0$, the original relation was given on a compact interval. Since we considered the space ${C}_{B}({\mathbb{R}}_{+})$, the inequality of Shisha and Mond holds on the unbounded interval ${\mathbb{R}}_{+}$. The proof of (17) is mainly based on relation

$$|f(x)-f(y)|\le \omega (f;|x-y|),$$ |

property (11) and Schwarz’ inequality.

Choosing $\mathrm{\Lambda}={L}_{n}$ and taking in view (5) and (9), relation (17) becomes

$$|({L}_{n}f)(x)-f(x)|\le \left(1+\frac{1}{\delta}\sqrt{{\mu}_{n,2}(x)}\right)\omega (f;\delta ),\delta >0.$$ | (18) |

On the other hand, from (9) we get

$${\mu}_{n,2}(x)\le {\lambda}_{n}^{2}{x}^{4}+\frac{x}{n{\lambda}_{n}}\le {\lambda}_{n}^{\ast}({x}^{4}+x).$$ | (19) |

Taking in (18) $\delta =\sqrt{{\lambda}_{n}^{\ast}}$, the proof is ended. $\mathrm{\square}$

Remarks. (i) If $f\in C({\mathbb{R}}_{+})$, considering the compact $[a,b]\subset {\mathbb{R}}_{+}$, the upper bound of the error of approximation indicated by (15) turns into

$${\Vert {L}_{n}f-f\Vert}_{\mathrm{\infty}}\le \left(1+\sqrt{{b}^{4}+b}\right){\omega}_{[a,b]}(f;{\lambda}_{n}^{\ast}),$$ |

where the norm is taken on the compact $[a,b]$.

(ii) In the light of the definition of ${\lambda}_{n}^{\ast}$ and relations (1), (3), we infer the following two possibilities, preserving the condition that the sequence ${({\lambda}_{n})}_{n\ge 1}$ to be strictly decreasing positive.

$\alpha )$ ${\lambda}_{n}\le {n}^{-1/3}$ and $\underset{n\to \mathrm{\infty}}{lim}n{\lambda}_{n}=\mathrm{\infty}$. In this case ${\lambda}_{n}^{\ast}={(n{\lambda}_{n})}^{-1}$.

$\beta )$ ${\lambda}_{n}>{n}^{-1/3}$ and $\underset{n\to \mathrm{\infty}}{lim}{\lambda}_{n}=0$. Now, we have ${\lambda}_{n}^{\ast}={\lambda}_{n}^{2}$.

It is noted that in the above, the numerical sequence ${({n}^{-1/3})}_{n\ge 1}$ appears. This sequence was used in the first study dedicated to Bernstein type rational functions, see K. Balázs’ work [7].

In the next step we prove a relation between the local smoothness of functions and the local approximation. For the completeness of the information, we recall that a continuous function $f:{\mathbb{R}}_{+}\to \mathbb{R}$ is locally $\mathrm{Lip}\alpha $ on $E$ ($$, $E\subset {\mathbb{R}}_{+}$) if it satisfies the condition

$$|f(x)-f(y)|\le {M}_{f}{|x-y|}^{\alpha},(x,y)\in {\mathbb{R}}_{+}\times E,$$ | (20) |

where ${M}_{f}$ is a constant depending only on $f$.

Also, the distance between $x\in {\mathbb{R}}_{+}$ and $E$ is denoted by $d(x,E)$ and defined as follows

$$d(x,E)=inf\{|x-y|:y\in E\}.$$ |

Theorem 3. Let ${L}_{n}$, $n\in \mathbb{N}$, be defined by (2), $$ and $E$ is a subset of ${\mathbb{R}}_{+}$. If $f$ is locally $\mathrm{Lip}\alpha $ on $E$, then

$$|({L}_{n}f)(x)-f(x)|\le {M}_{f}({({\lambda}_{n}^{\ast})}^{\alpha /2}\mathrm{max}\{{x}^{2\alpha},{x}^{\alpha /2}\}+2{d}^{\alpha}(x,E)),x\ge 0,$$ |

takes place, where ${\lambda}_{n}^{\ast}$ is given by (15).

Proof. At the beginning we recall a classical relation stemming from Hölder’s inequality. Considering $r=2{\alpha}^{-1}$ in the relation $1/r+1/s=1$, $r>0$, $s>0$, from (2) and (5) we obtain

$${L}_{n}({h}^{\alpha};x)\le {({L}_{n}({h}^{2};x))}^{\alpha /2},x\ge 0,$$ |

where $\alpha \in (0,1]$ and $h\ge 0$. Choosing $h=|{\phi}_{x,1}|$, see (8), and using (19), we get

$${L}_{n}({|{e}_{1}-x{e}_{0}|}^{\alpha};x)\le {\mu}_{n,2}^{\alpha /2}(x)\le {({\lambda}_{n}^{\ast}({x}^{4}+x))}^{\alpha /2},x\ge 0.$$ | (21) |

By using the continuity of $f$, it is obvious that (20) holds for any $x\ge 0$ and $y\in \overline{E}$, $\overline{E}$ being the closure of the set $E$. Let $(x,{x}_{0})\in {\mathbb{R}}_{+}\times \overline{E}$ be such that $|x-{x}_{0}|=d(x,E)$. We can write

$$|f-f(x)|\le |f-f({x}_{0})|+|f({x}_{0})-f(x)|$$ |

and applying the operators ${L}_{n}$ (linear and positive, consequently monotone), we have

$|({L}_{n}f)(x)-f(x)|$ | $\le {L}_{n}(|f-f({x}_{0})|;x)+|f(x)-f({x}_{0})|$ | |||

$\le {L}_{n}({M}_{f}{|{e}_{1}-{x}_{0}{e}_{0}|}^{\alpha};x)+{M}_{f}{(x-{x}_{0})}^{\alpha}.$ | (22) |

In the well known inequality ${(a+b)}^{\alpha}\le {a}^{\alpha}+{b}^{\alpha}$ ($a\ge 0$, $b\ge 0$, $$), we take $a=|{e}_{1}-x{e}_{0}|$, $b=|x-{x}_{0}|{e}_{0}$, consequently

$${|{e}_{1}-{x}_{0}{e}_{0}|}^{\alpha}\le {|{e}_{1}-x{e}_{0}|}^{\alpha}+{(|x-{x}_{0}|{e}_{0})}^{\alpha}.$$ |

By using (5) and (21) we can write

${L}_{n}({M}_{f}{|{e}_{1}-{x}_{0}{e}_{0}|}^{\alpha};x)$ | $\le {M}_{f}({L}_{n}({|{e}_{1}-x{e}_{0}|}^{\alpha};x)+{|x-{x}_{0}|}^{\alpha})$ | ||

$\le {M}_{f}({({\lambda}_{n}^{\ast}{x}^{4})}^{\alpha /2}+{({\lambda}_{n}^{\ast}x)}^{\alpha /2}+{|x-{x}_{0}|}^{\alpha}).$ |

Returning at (3), the conclusion of our theorem is proved. $\mathrm{\square}$

Unfortunately this class of operators is not an approximation process in some weighted spaces. We present such a negative result, indicating first the general frame of our study. Let $\phi $ be a continuous and strictly increasing function defined on ${\mathbb{R}}_{+}$ such that $\underset{x\to \mathrm{\infty}}{lim}\phi (x)=\mathrm{\infty}$. Set $\rho =1+{\phi}^{2}$. We consider the weighted space

$${B}_{\rho}({\mathbb{R}}_{+})=\{f:{\mathbb{R}}_{+}\to \mathbb{R}\mid |f(x)|\le {M}_{f}\rho (x),x\ge 0\},$$ |

${M}_{f}$ being a positive constant depending only on $f$, and the following subspaces of ${B}_{\rho}({\mathbb{R}}_{+})$

$${C}_{\rho}({\mathbb{R}}_{+})=C({\mathbb{R}}_{+})\cap {B}_{\rho}({\mathbb{R}}_{+}),$$ |

$${C}_{\rho}^{0}({\mathbb{R}}_{+})=\{f\in {C}_{\rho}({\mathbb{R}}_{+})\mid \underset{x\to \mathrm{\infty}}{lim}\frac{f(x)}{\rho (x)}\text{exists and is finite}\}.$$ |

The usual norm of these spaces is $\parallel \cdot {\parallel}_{\rho}$ defined by

$${\Vert f\Vert}_{\rho}=\underset{x\ge 0}{sup}\frac{|f(x)|}{\rho (x)}.$$ | (23) |

For any $f\in {C}_{\rho}({\mathbb{R}}_{+})$, based on (23), we have

$$|({L}_{n}f)(x)|\le {\Vert f\Vert}_{\rho}({L}_{n}\rho )(x),n\ge 1.$$ |

Hence, the operator ${L}_{n}$ maps ${C}_{\rho}({\mathbb{R}}_{+})$ into ${C}_{\rho}({\mathbb{R}}_{+})\subset {B}_{\rho}({\mathbb{R}}_{+})$ if and only if

$$({L}_{n}\rho )(x)\le M\rho (x),x\ge 0,$$ | (24) |

where $M$ is a constant.

We recall the following result due to Gadzhiev [12, Theorem 2] which says: a sequence ${({A}_{n})}_{n\ge 1}$ of linear and positive operators acting from ${C}_{\rho}({\mathbb{R}}_{+})$ to ${B}_{\rho}({\mathbb{R}}_{+})$ satisfies

$$\underset{n\to \mathrm{\infty}}{lim}{\Vert {A}_{n}f-f\Vert}_{\rho}=0\text{for every}f\in {C}_{\rho}^{0}({\mathbb{R}}_{+}),$$ | (25) |

if and only if

$$\underset{n\to \mathrm{\infty}}{lim}{\Vert {A}_{n}{\phi}^{\nu}-{\phi}^{\nu}\Vert}_{\rho}=0,\nu =0,1,2.$$ | (26) |

We mention that in [12] the result was presented without ”only if” and considering the set $\mathbb{R}$. Clearly, to insert ”only if” is trivial and the use of the set ${\mathbb{R}}_{+}$ does not change the result.

As it seen above, the weight function $\rho $ not only characterizes the growth of $f$ at infinity but also defines the test functions in Korovkin type theorem.

Choosing a very simple weight namely $\phi :={e}_{1}$, based on (5) and (7), we find that for ${L}_{n}$, $n\ge 1$, relation (24) is fulfilled. For example, we can consider $M=1+{(\underset{n\ge 1}{inf}(n{\lambda}_{n}))}^{-1}$. Trying to apply Gadzhiev’s result for our operators, we notice that (26) does not hold for $\nu =2$. Indeed,

$$\underset{x\ge 0}{sup}\frac{|({L}_{n}{e}_{2})(x)-{x}^{2}|}{1+{x}^{2}}=\underset{x\ge 0}{sup}\frac{|n{\lambda}_{n}^{3}{x}^{4}+2n{\lambda}_{n}^{2}{x}^{3}-x|}{n{\lambda}_{n}(1+{x}^{2}){(1+{\lambda}_{n}x)}^{2}}\ge 1,$$ |

consequently $\underset{n\to \mathrm{\infty}}{lim}{\Vert {L}_{n}{e}_{2}-{e}_{2}\Vert}_{\rho}\ne 0$.

We can enunciate

Theorem 4. The sequence ${({L}_{n})}_{n\ge 1}$ defined by (2) is not an approximation process in the space ${C}_{\rho}^{0}({\mathbb{R}}_{+})$, where $\rho =1+{e}_{2}$.

It is worth mentioning that for this particular weight we could use the fact that $\{{e}_{0},{e}_{1},{e}_{2}\}$ is a strict Korovkin subset in ${C}_{\rho}^{0}({\mathbb{R}}_{+})$, see the monograph [6, Proposition 4.2.5, statement (6)].

For a more general weight, i.e. $\phi (x)={x}^{m/2}$, $m\ge 2$ fixed, we give estimates of the error $|({L}_{n}f)(x)-f(x)|$, $n\in \mathbb{N}$, for functions $f$ belonging to ${C}_{\rho}^{0}({\mathbb{R}}_{+})$. Clearly, this does not mean that the sequence is an approximation process on the mentioned space, the evaluation being pointwise. For our purpose we use a weighted modulus of smoothness defined as follows

$$ | (27) |

Obviously, ${\mathrm{\Omega}}_{m}(f;\delta )\le 2{\Vert f\Vert}_{\rho}$, $\delta >0$, $f\in {B}_{\rho}({\mathbb{R}}_{+})$, and it possesses the following main properties ([15])

$$\{\begin{array}{ccc}{\mathrm{\Omega}}_{m}(f;\cdot )\text{is a monotone increasing function},\hfill & & \\ {\mathrm{\Omega}}_{m}(f;\lambda \delta )\le (\lambda +1){\mathrm{\Omega}}_{m}(f;\delta ),\delta 0,\lambda 0,\hfill & & \\ \underset{\delta \to {0}^{+}}{lim}{\mathrm{\Omega}}_{m}(f;\delta )=0,\text{for any}f\in {C}_{\rho}^{0}({\mathbb{R}}_{+}).\hfill & & \end{array}$$ | (28) |

Theorem 5. Let the operators ${L}_{n}$, $n\in \mathbb{N}$, be defined by (2) such that the requirements (1) and (3) are fulfilled. For any function $f\in {C}_{\rho}^{0}({\mathbb{R}}_{+})$, $\rho =1+{e}_{m}$ with $m\ge 2$, the following inequality

$$|({L}_{n}f)(x)-f(x)|\le C(1+{x}^{m})\sqrt{{x}^{4}+x}{\mathrm{\Omega}}_{m}(f;\sqrt{{\lambda}_{n}^{\ast}})$$ | (29) |

takes place, where $C$ is a constant depending only on $m$, ${\mathrm{\Omega}}_{m}(f;\cdot )$ and ${\lambda}_{n}^{\ast}$ are defined by (27) and (16), respectively.

Proof. For $x=0$ the statement is evident. Let $n\in \mathbb{N}$, $f\in {C}_{\rho}^{0}({\mathbb{R}}_{+})$ and $x>0$ be arbitrarily fixed.

Set ${\theta}_{x,m}(t)=1+{(2x+t)}^{m}$. We recall $|{\phi}_{x,1}(t)|=|t-x|$, see (8). By using definition of ${\mathrm{\Omega}}_{m}(f;\cdot )$ and properties (28), we can write

$|f(t)-f(x)|$ | $\le (1+{(x+|t-x|)}^{m})\left(1+{\displaystyle \frac{1}{\delta}}|t-x|\right){\mathrm{\Omega}}_{m}(f;\delta )$ | ||

$\le {\theta}_{x,m}(t)\left(1+{\displaystyle \frac{1}{\delta}}|{\phi}_{x,1}(t)|\right){\mathrm{\Omega}}_{m}(f;\delta ).$ |

Knowing that the operators are linear positive therefore monotone, and applying Cauchy inequality, we get

$$|({L}_{n}f)(x)-f(x)|\le {L}_{n}(|f-f(x)|;x)$$ |

$$\le \left(({L}_{n}{\theta}_{x,m})(x)+\frac{1}{\delta}\sqrt{({L}_{n}{\theta}_{x,m}^{2})(x)}\sqrt{({L}_{n}{\phi}_{x,1}^{2})(x)}\right){\mathrm{\Omega}}_{m}(f;\delta ).$$ | (30) |

Since ${L}_{n}{e}_{m}\le {C}_{1}(1+{e}_{m})$, ${C}_{1}$ a certain constant depending on $m$, there are positive constants ${C}_{2}$ and ${C}_{3}$ depending also on $m$, such that

$$({L}_{n}{\theta}_{x,m})(x)\le {C}_{2}(1+{x}^{m})\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{(({L}_{n}{\theta}_{x,m}^{2})(x))}^{1/2}\le {C}_{3}(1+{x}^{m}).$$ |

On the other hand, based on (9), we deduce

$$({L}_{n}{\phi}_{x,1}^{2})(x)={\mu}_{n,2}(x)\le {\lambda}_{n}^{\ast}({x}^{4}+x).$$ |

Returning at (30) and choosing $\delta =\sqrt{{\lambda}_{n}^{\ast}}$, the result follows. $\mathrm{\square}$

Remark. In the following, for $m\ge 2$ fixed, we set ${\rho}_{1}=1+{e}_{m}$ and ${\rho}_{2}=1+{e}_{m+2}$. Clearly, ${B}_{{\rho}_{1}}({\mathbb{R}}_{+})\subset {B}_{{\rho}_{2}}({\mathbb{R}}_{+})$. Taking in view relation (29) we can write

$${\Vert {L}_{n}f-f\Vert}_{{\rho}_{2}}\le \stackrel{~}{C}{\mathrm{\Omega}}_{m}(f;\sqrt{{\lambda}_{n}^{\ast}}),$$ |

for every $f\in {C}_{{\rho}_{1}}({\mathbb{R}}_{+})$, where $$. Due to the assumptions (1) and (3), we obtain $\underset{n\to \mathrm{\infty}}{lim}{\lambda}_{n}^{\ast}=0$ and the last property of ${\mathrm{\Omega}}_{m}(f;\cdot )$ at (28) implies

$$\underset{n\to \mathrm{\infty}}{lim}{\Vert {L}_{n}f-f\Vert}_{{\rho}_{2}}=0,f\in {C}_{{\rho}_{1}}^{0}({\mathbb{R}}_{+}).$$ |

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[1] Balazs, K. (1975). Approximation by Bernstein type rational functions. Acta Math. Acad. Sci. Hungar. 26(1–2):123–134. DOI: 10.1007/BF01895955.

[2] Balazs, C., Szabados, J. (1982). Approximation by Bernstein type rational functions. II. Acta Math. Acad. Sci. Hungar. 40(3–4):331–337. DOI: 10.1007/BF01903593.

[3] Totik, V. (1984). Saturation for Bernstein type rational functions. Acta Math. Hung. Hungar. 43(3–4):219–250. DOI: 10.1007/BF01958021.

[4] Vecchia, B. D. (1989). On some preservation and asymptotic relations of a rational operators. Facta Universitatis (Nis). 4:57–62.

[5] Abel, U., Della Vecchia, B. (2000). Asymptotic approximation by the operators of K. Balazs and Szabados. Acta Sci. Math. (Szeged). 66:137–145.

[6] Agratini, O. (2002). On approximation properties of Balazs-Szabados operators and their Kantorovich extension. Korean J. Comput. Appl. Math. 9(2):361–372.

[7] Dogru, O. (2006). On statistical approximation properties of Stancu type bivariate generalization of q-Balazs-Szabados operators. In: Octavian A., Petru, B., eds. Numerical Analysis and Approximation Theory. Cluj-Napoca: Casa Cartii de Stiinta, pp. 179–194. Paper presented at the Proceedings of the International Conference (NAAT 2006) held at Babes-Bolyai University, Cluj-Napoca, July 5–8, 2006.

[8] Gal, S. G. (2009). Approximation by Complex Bernstein and Convolution Type Operators, Series on Concrete and Applicable Mathematics, Vol. 8. Hackensack, NJ: World Scientific Publishing.

[9] Ispir, N., Yildiz Ozkan, E. (2013). Approximation properties of complex q-Balazs-Szabados operators in compact disks. J. Inequal. Appl. 2013:361. DOI: 10.1186/1029-242X-2013-361.

[10] Mahmudov, N. I. (2016). Approximation properties of the q-Balazs-Szabados complex operators in the case q 1: Comput. Meth. Funct. Theory Theory. 16(4):567–583. Issue DOI: 10.1007/s40315-015-0154-7.

[11] Abel, U., Agratini, O. (2016). On the variation detracting property of operators of Balazs and Szabados. Acta Math. Hungar. Hungar. 150(2):383–395. Issue DOI: 10.1007/s10474-016-0642-x.

[12] Abdulofizov, S. (1980). Approximation by rational functions of the type of generalized Bernstein polynomials on the semiaxis. Izv. Akad. Nauk Tadzhik. SSR, Otdel. Fiz.-Mat. Khim. i Geol. Nauk. 2(76):78–81. (in Russian, Tajiki summary).

[13] Abdulofizov, S., Videnskii, V. S. (1978). Approximation on a semi-axis by rational functions of Bernstein polynomial type. In: 31st Gertsen Lectures: Non-Linear Functional Analysis and the Theory of Approximation of Functions. Leningrad: Leningrad, Gos. Ped. Inst, pp. 1–3 (in Russian).

[14] Lorentz, G. G. (1986). Bernstein Polynomials, 2nd ed. New York, NY: Chelsea Publ. Comp.

[15] DeVore, R. A., Lorentz, G. G. (1993). Constructive approximation. In: Grundlehren Der Mathematischen Wissenschaften. Berlin, Heidelberg: Springer-Verlag, pp. 303.

[16] Altomare, F., Campiti, M. (1994). Korovkin-Type Approximation Theory and Its Applications, Walter de Gruyter Studies in Mathematics, Vol. 17. Berlin: de Gruyter & Co.

[17] Shisha, O., Mond, B. (1968). The degree of convergence of sequences of linear positive operators. Proc. Natl. Acad. Sci. USA. 60(4):1196–1200. DOI: 10.1073/pnas.60.4.1196.

[18] Gadzhiev, A. D. (1976). Theorems of Korovkin type. Math. Notes 20(5):995–998. DOI: 10.1007/BF01146928.

[19] Lopez-Moreno, A.-J. (2004). Weighted simultaneous approximation with Baskakov type operators. Acta Mathematica Hungar. 104(1/2):143–151. DOI: 10.1023/B:AMHU.0000034368.81211.23.