This note falls under the field of Approximation Theory, more precisely it aims at the study of linear and positive approximation processes. It is known that Szász operators, along with the many discrete and integral generalizations obtained over time, play a significant role in this domain.

Our present study concerns a class of Szász-Durrmeyer type operators introduced by Păltănea in 2008 [9]. They depend on two parameters and are described as follows. For $\alpha >0$, $\rho >0$ and $x\in {ℝ}_{+}=[0,\mathrm{\infty })$,

where

and $f:{ℝ}_{+}\to ℝ$ is a locally integrable function for which formula (1) is well defined for all $x\ge 0$.

Păltănea operators preserve affine functions and make a link between the Phillips operators and the classical Szász operators. In the particular case $\rho =1$ and $\alpha =n\in \mathbb{N}$, the above operators become Phillips operators [11]

In the limit case $\rho \to \mathrm{\infty }$ (see Theorem 2) one obtains the classical Szász operators defined by

The extension in the Durrmeyer sense of $S_n$ operators was achieved by Mazhar and Totik [7] in 1985. Unlike Păltănea operators, this extension does not reproduce affine functions.

Our goal is twofold: to collect some known properties of Păltănea’s operators in a brief synthesis as well as to construct a version of King type by investigating its utility and its approximation properties. Wanting to create a self-contained presentation, all the notions used are described explicitly. As much as possible, we have kept the original notations used in the papers cited.

We did not propose an exhaustive presentation of the results obtained over time, but scoring of the most significant properties of this approximation process. Păltănea studied this class of operators in the papers [9], [10], the following main properties being proved.

Set $W$, the space of functions $f:{ℝ}_{+}\to ℝ$ which are Riemann integrable on each compact interval of ${ℝ}_{+}$ and for which exist certain numbers $M>0$, $q>0$ such that $|f(t)|\le M{e}^{qt}$, $t\ge 0$. For $\alpha >0$ and $\rho >0$, denote by $W_{\alpha }^{\rho }$ the subspace of $W$ of those functions $f$ which satisfy the above inequality with . One has

In [9, Theorem 2.1] it was showed that $L_{\alpha }^{\rho }f$ exists for any $f\in W_{\alpha }^{\rho }$, $\alpha >0$, $\rho >0$.

An approximation property in the particular case $\rho =n\in \mathbb{N}$ will be read as follows.

The limit of the functions $L_{\alpha }^{\rho }f$ has been established when $\rho$ tends to infinity.

Also it was proved that $L_{\alpha }^{\rho }$ preserves convexity of higher order and it has the property of simultaneous approximation on the compact sets [10, Theorems 6, 9].

Further, we will highlight some papers based on Păltănea operators and which bear his name in the title.

We recall that in [5] the authors gave a generalization of Szász operators based on Appell polynomials. Let $g(z)={\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{a}_k{z}^k$ be an analytic function in the disc , $R>1$, and $g(1)\ne 0$. The Appell polynomials $p_k$, $k\in {\mathbb{N}}_0=\{0\}\cup \mathbb{N}$, are defined by the generating function

For $f\in C({ℝ}_{+})$, Verma and Gupta [14] proposed the Jakimovski-Leviatan-Păltănea operators defined as

where ${\mathrm{\Theta }}_{n,k}^{\rho }$ is defined at (1) and

$p_k$ being described at (4).

To establish the rate of convergence, the authors used the moduli of smoothness of the first and second order which give direct information about the smoothness of $f$. We recall their definitions.

where $f\in C_B({ℝ}_{+})$, the space of continuous and bounded real valued functions defined on ${ℝ}_{+}$. The following result was proved.

Considering the space

$M_f$ being a constant depending on $f$, a Voronovskaja type asymptotic formula was also obtained.

Goyal and Agrawal [3] defined and studied the Bézier variant of the operators $M_{n,\rho }$, $n\in \mathbb{N}$.

Set $C_{\gamma }({ℝ}_{+})=\{f\in C({ℝ}_{+}):f(t)=𝒪(e^{\gamma t})\text{ as }t\to \mathrm{\infty }\}$, where $\gamma >0$ is fixed. For $\theta \ge 1$ and $f\in C_{\gamma }({ℝ}_{+})$, the Jakimovski-Leviatan-Păltănea-Bézier operator is of the form

where

see [3, Eq. (1.2)].

Clearly, $J_{n,k}(x)-J_{n,k+1}(x)=l_{n,k}(x)$ defined by (6), $k\in {\mathbb{N}}_0$. For $\theta =1$, $M_{n,\rho }^1$ turns out to be $M_{n,\rho }$ defined by (5). A substantial result is the establishment of the rate of convergence for functions having a derivative of bounded variation.

Let $DB{V}_{\gamma }({ℝ}_{+}^{\ast })$, $\gamma \ge 0$, be the class of all functions defined on ${ℝ}_{+}^{\ast }$ having a derivative of bounded variation on every bounded subinterval of ${ℝ}_{+}^{\ast }$ and any function $f$ of this class enjoys the property $|f(t)|\le M{t}^{\gamma }$, $t\in {ℝ}_{+}^{\ast }$.

Motivated by the above mentioned construction, in [8] the authors introduced the Bézier-Păltănea operators based on Gould-Hopper polynomials. For this new generalization of Păltănea operators, the authors obtained both the quantitative Voronovskaja type theorem in terms of Ditzian-Totik modulus of smoothness and the rate of pointwise convergence for the functions having a derivative of bounded variation.

In the final part we mention a recent result obtained by Gupta and Agrawal [4]. They proposed a hybrid integral type operator containing both Szász as well as Baskakov bases in summation. More precisely, in (1) they replaced $s_{\alpha ,k}(x)$, $k\in {\mathbb{N}}_0$, $x\in {ℝ}_{+}$, with $p_{\alpha ,k}(x,c)$, where

$c$ being a constant belonging to the interval $(0,1]$. In the above ${(\alpha /c)}_k$ stands for rising factorial, also called Pochhammer function. We recall ${(\alpha /c)}_0$ is taken to be 1. For these new operators, the notation $B_{\alpha }^{\rho }(f;\cdot ,c)$ was used. Among the results obtained we mention a Grüss-Voronovskaja type theorem. Setting

where $w(x)=1+x^2$, the following statement was proved.

Set $e_0(x)=1$, $e_j(x)=x^j$ $(j\in \mathbb{N})$, $x\ge 0$.

Two decades ago, King [6] had the idea to modify the Bernstein operators such that to reproduce the monomials $e_0$ and $e_2$. Consequently, the modified operators enjoy the property of keeping the functions $c_1{e}_0+c_2{e}_2$ as fixed points, for any real constants $c_1$ and $c_2$.

From approximation theory point of view the construction is useful. In spite of the fact that the new operators have the degree of exactness null, the maximum rate of convergence is smaller. Over time this technique was applied to many linear approximation processes, becoming known as the King method. We propose to apply it to Păltănea operators (1). It is known that

see [10, Eq. (2.1)]. Considering

where $\beta (\alpha ,\rho )=(\rho +1)/(\alpha \rho )$, we define the operators

$f\in W$.

In the above $\delta$ represents Dirac delta function for which

(ii) For any $f\in C_B({ℝ}_{+})$ we can easily deduce that the operators are non-expansive, this means $\Vert L_{\alpha ,\rho }^{\ast }f\Vert \le \Vert f\Vert$. The proof uses the identities

Relations (10) and (11) involve the identities

Since any compact interval $K\subset {ℝ}_{+}$ is isomorphic to $[0,b]$, $b>0$ arbitrarily fixed, in our approach will use only this interval.

Let $b>0$ be arbitrarily fixed. The relation $\underset{\alpha \to \mathrm{\infty }}{lim}u(x)=x$ uniformly on $[0,b]$ takes place. Based on (14), the proof of the above theorem follows exactly the same line as the proof of Theorem 3.4 from [9], so we omit it.

For a positive linear operator $\mathrm{\Lambda }$ its second central moment defined by

where

plays a crucial role when estimating its local rate of convergence.

The identities (10) and (14) imply

It turns out that the second central moment of their King type variant (12) is smaller than the second central moment of the Păltănea operators (1) on the whole interval $(0,+\mathrm{\infty })$.

By virtue of the classical results regarding the local rate of convergence established by Shisha and Mond [12], the relations (10), (14) and (16) guarantee

for any $f\in C_B({ℝ}_{+})$, where ${\omega }_1$ is defined at (7).

The evaluation of the rate of convergence can be carried out in weighted spaces, for example in $C_2^{\ast }({ℝ}_{+})$ defined at (9) and endowed with the usual norm $\parallel \cdot {\parallel }_{C_2^{\ast }({ℝ}_{+})}$,

where $w(x)=1+x^2$, $x\ge 0$.

To obtain the following new result we need an inequality that we present in what follows. Any discrete or integral linear positive operator $\mathrm{\Lambda }$ of summation type satisfies the classical inequality

where ${\phi }_x$ is given at (15). Because the operators $L_{\alpha ,\rho }^{\ast }$ contain as the first term a quantity not included in the sum, for a self-contained presentation, we prove the relation

The proof is based on Cauchy–Schwarz inequality both for integrals and for series, and it runs as follows.

see (13).

Further, we define $b_0={\phi }_x^2(0)=x^2$ and $b_k={\displaystyle {\int }_0^{\mathrm{\infty }}}{\mathrm{\Theta }}_{\alpha ,k}^{\rho }(t){\phi }_x^2(t)𝑑t,k\ge 1$. We get

and the proof of (18) is completed.

A less frequently used tool to approximate signals is the so called Steklov mean. The benefit of special function is that continuous functions can be approximated by smoother functions. For $f\in C_B({ℝ}_{+})$, the Stelov mean of second order and step $h/2$ is defined by

and verifies the inequalities

and if $f_h^{\prime },f_h^{\prime \prime }\in C_B({ℝ}_{+})$ exist,

In the above $\parallel \cdot \parallel$ stands for the sup-norm, $\Vert h\Vert =\underset{x\ge 0}{sup}|h(x)|$, $h\in C_B({ℝ}_{+})$.

The key of the proofs of these relations consists in rewriting the definitions (7) and (8) as follows

where $\delta \ge 0$. The proofs of (20) and (21) can be found in [2, Eqs. (5.2)-(5.4)].