This note concerns the study of an approximation linear positive process introduced by R. Paltanea in 2008. Considering the impact of this class of operators that depends on two parameters, in a distinct section we present a brief radiograph of the main properties highlighted over time in various papers. Our contribution materializes in the definition and study of the approximation properties of the King variant of these operators.


U. Abel

O. Agratini
Tiberiu Popoviciu Institute of Numerical Analysis Romanian Academy, Romania


linear approximation process, Durrmeyer type operators, modulus of smoothness, King type operator.

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U. Abel, O. Agratini, Păltănea’s operators: old and new results, Bulletin of the Transilvania University of Brașov, Serie III: Mathematics and Computer Science,  3(65) (2023) no. 2, pp. 1.12, https://doi.org/10.31926/but.mif.2023.


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Bulletin of the Transilvania University of Brasov, Series III: Mathematics and Computer Science

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Păltănea’s operators: old and new results

Păltănea’s operators: old and new results

Ulrich Abel Fachbereich MND
Technische Hochschule Mittelhessen
Wilhelm-Leuschner-Straße 13
61169 Friedberg, Germany
ORCID: 0000-0003-1889-4850
 and  Octavian Agratini Tiberiu Popoviciu Institute of Numerical Analysis
Romanian Academy
Str. Fântânele, 57, 400320 Cluj-Napoca, Romania
ORCID: 0000-0002-2406-4274

This note concerns the study of an approximation linear positive process introduced by R. Păltănea in 2008. Considering the impact of this class of operators that depends on two parameters, in a distinct section we realize a brief radiograph of the main properties highlighted over time in various papers. Our contribution materialized in the definition and study of the approximation properties of King variant of these operators.

Mathematics Subject Classification: 41A36, 41A35, 41A25.

Keywords and phrases: linear approximation process, Durrmeyer type operators, modulus of smoothness, King type operator.

1. Introduction

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 α>0, ρ>0 and x+=[0,),

Lαρ(f;x)=eαxf(0)+k=1sα,k(x)0Θα,kρ(t)f(t)𝑑t, (1)


Θα,kρ(t)=αρΓ(kρ)eαρt(αρt)kρ1 (2)

and f:+ is a locally integrable function for which formula (1) is well defined for all x0.

Păltănea operators preserve affine functions and make a link between the Phillips operators and the classical Szász operators. In the particular case ρ=1 and α=n, the above operators become Phillips operators [11]


In the limit case ρ (see Theorem 2) one obtains the classical Szász operators defined by

Sn(f;x)=k=0sn,k(x)f(kn),x0. (3)

The extension in the Durrmeyer sense of Sn 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.

2. An eclectic collection of known results

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:+ which are Riemann integrable on each compact interval of + and for which exist certain numbers M>0, q>0 such that |f(t)|Meqt, t0. For α>0 and ρ>0, denote by Wαρ the subspace of W of those functions f which satisfy the above inequality with q<αρ. One has


In [9, Theorem 2.1] it was showed that Lαρf exists for any fWαρ, α>0, ρ>0.

An approximation property in the particular case ρ=n will be read as follows.

Theorem 1.

([9, Theorem 3.4]). For any function fWC(+) and any number n, there exists α0>0 such that Lαnf exists for α>α0 and for any compact set K+ we have

limαLαnf=f, uniformly on K.

The limit of the functions Lαρf has been established when ρ tends to infinity.

Theorem 2.

([10, Theorem 4]). For any α>0, any fW and any b>0, there is ρ0>0 such that Lαρf exists for all ρρ0 and we have

limρLαρ(f;x)=Sα(f;x), uniformly for x[0,b],

where the Szász operator Sαf is defined in (3).

Also it was proved that Lαρ 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)=k=0akzk be an analytic function in the disc |z|<R, R>1, and g(1)0. The Appell polynomials pk, k0={0}, are defined by the generating function

g(u)eux=k=0pk(x)uk. (4)

For fC(+), Verma and Gupta [14] proposed the Jakimovski-Leviatan-Păltănea operators defined as

Mn,ρ(f;x)=ln,0(x)f(0)+k=1ln,k(x)0Θn,kρ(t)f(t)𝑑t, (5)

where Θn,kρ is defined at (1) and

ln,k(x)=enxg(1)pk(nx),k0, (6)

pk 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.

ω1(f,δ)=sup0hδsupx0|f(x+h)f(x)|, (7)
ω2(f,δ)=sup0hδsupx0|f(x+2h)2f(x+h)+f(x)|, (8)

where fCB(+), the space of continuous and bounded real valued functions defined on +. The following result was proved.

Theorem 3.

([14, Theorem 2]). For fCB(+), we have


where δ=(Mn,ρ((x)2;x)+(g(1)ng(1))2)1/2.

Considering the space


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

Theorem 4.

([14, Theorem 3]). For any function fBw(+)C(+) such that f, f′′ are continuous and belong to Bw(+), we have


Goyal and Agrawal [3] defined and studied the Bézier variant of the operators Mn,ρ, n.

Set Cγ(+)={fC(+):f(t)=𝒪(eγt) as t}, where γ>0 is fixed. For θ1 and fCγ(+), the Jakimovski-Leviatan-Păltănea-Bézier operator is of the form




see [3, Eq. (1.2)].

Clearly, Jn,k(x)Jn,k+1(x)=ln,k(x) defined by (6), k0. For θ=1, Mn,ρ1 turns out to be Mn,ρ defined by (5). A substantial result is the establishment of the rate of convergence for functions having a derivative of bounded variation.

Let DBVγ(+), γ0, be the class of all functions defined on + having a derivative of bounded variation on every bounded subinterval of + and any function f of this class enjoys the property |f(t)|Mtγ, t+.

Theorem 5.

([3, Theorem 5]). Let fDBVγ(+), θ1 and let Vcd(fx) be the total variation of fx on [c,d]+. For every x+ and sufficiently large n, we have

|Mn,ρθ(f;x)f(x)| θθ+1Cx(1+ρ)nρ|f(x+)+θf(x)|

where C>1 and the function fx is defined by


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 sα,k(x), k0, x+, with pα,k(x,c), where


c being a constant belonging to the interval (0,1]. In the above (α/c)k stands for rising factorial, also called Pochhammer function. We recall (α/c)0 is taken to be 1. For these new operators, the notation Bαρ(f;,c) was used. Among the results obtained we mention a Grüss-Voronovskaja type theorem. Setting

C2(+)={fBw(+)C(+):limxf(x)w(x) exists and is finite}, (9)

where w(x)=1+x2, the following statement was proved.

Theorem 6.

([4, Theorem 3.2]). Let f,g,f,g,f′′,g′′,(fg),(fg)′′ belong to C2(+). For any x+ we have


3. On the King variant of the operators Lαρ

Set e0(x)=1, ej(x)=xj (j), x0.

Two decades ago, King [6] had the idea to modify the Bernstein operators such that to reproduce the monomials e0 and e2. Consequently, the modified operators enjoy the property of keeping the functions c1e0+c2e2 as fixed points, for any real constants c1 and c2.

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

Lαρe0=e0,Lαρe1=e1,Lαρe2=e2+ρ+1αρe1,α>0,ρ>0, (10)

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

u(x)=12(β2(α,ρ)+4x2β(α,ρ)),x0, (11)

where β(α,ρ)=(ρ+1)/(αρ), we define the operators

Lα,ρ(f;x)=eαu(x)f(0)+k=1sα,k(u(x))0Θα,kρ(t)f(t)𝑑t,x0, (12)


Remarks. (i) By using a bivariate kernel, we can write (12) in a more compact form, as follows




In the above δ represents Dirac delta function for which


(ii) For any fCB(+) we can easily deduce that the operators are non-expansive, this means Lα,ρff. The proof uses the identities

0Θα,kρ(t)𝑑t=1,k. (13)

Relations (10) and (11) involve the identities

Lα,ρe0=e0,Lα,ρe1=u,Lα,ρe2=e2,α>0,ρ>0. (14)

Since any compact interval K+ is isomorphic to [0,b], b>0 arbitrarily fixed, in our approach will use only this interval.

Theorem 7.

For any b>0, any function fWC(+) and any number n, there exists α0>0 such that for α>α0, Lα,n is well-defined and we have

limαLα,nf=f, uniformly on [0,b].

Let b>0 be arbitrarily fixed. The relation limαu(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 Λ its second central moment defined by



φx(t)=tx,(t,x)+×+ (15)

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

The identities (10) and (14) imply

μ2(Lαρ;x)=β(α,ρ)x,μ2(Lα,ρ;x)=2x(xu(x)). (16)

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,+).

Lemma 1.

(i) For x>0, there holds


(ii) The inequality


is valid, for each x>0.


(i) Let x>0. The first inequality follows from the observation


since β(α,ρ)>0. Furthermore, we have


which proves the second inequality.

(ii) By (16), this statement is a consequence of the previous result. ∎

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 fCB(+), where ω1 is defined at (7).

Remark. Since ω1 associated with a function f is an increasing function, Lemma 1 (ii) demonstrates that the upper bound for the absolute error of Lα,ρ is smaller than that for Lαρ.

The evaluation of the rate of convergence can be carried out in weighted spaces, for example in C2(+) defined at (9) and endowed with the usual norm C2(+),


where w(x)=1+x2, x0.

Theorem 8.

Let Lα,ρ be defined by (12). For every fWC2(+), Lα,ρ converges to f in norm, i.e.,

limαLα,ρffC2(+)=0. (17)

It is known that {e0,e1,e2} is a Korovkin set in C2(+), see, e.g., [1, Proposition 4.2.5.-(6)]. Taking in view identities (14), it remains for us to prove (17) only for f:=e1. Applying two times Lemma 1 (i), we obtain the estimate


for all x0, from which we deduce


Thus, we got what we proposed, consequently (17) takes place. ∎

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


where φx is given at (15). Because the operators Lα,ρ contain as the first term a quantity not included in the sum, for a self-contained presentation, we prove the relation

Lα,ρ|φx|(Lα,ρφx2)1/2,x0. (18)

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

0Θα,kρ(t)|φx|(t)𝑑t (0Θα,kρ(t)𝑑t)1/2(0Θα,kρ(t)φx2(t)𝑑t)1/2

see (13).

Further, we define b0=φx2(0)=x2 and bk=0Θα,kρ(t)φx2(t)𝑑t,k1. We get

(Lα,ρ|φx|)(x) k=0sα,k(u(x))bk1/2=k=0sα,k(u(x))sα,k(u(x))bk1/2

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 fCB(+), the Stelov mean of second order and step h/2 is defined by

fh(x)=4h20h/20h/2(2f(x+u+v)f(x+2(u+v)))𝑑u𝑑v (19)

and verifies the inequalities

fhfω2(f,h), (20)

and if fh,fh′′CB(+) exist,

fh5hω1(f,h),fh′′9h2ω2(f,h). (21)

In the above stands for the sup-norm, h=supx0|h(x)|, hCB(+).

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


where δ0. The proofs of (20) and (21) can be found in [2, Eqs. (5.2)-(5.4)].

Remark. For the full information of the reader, in accordance with The Great Soviet Encyclopedia, 3rd Edition (1969-1978), we mention that the initial form of this type of function was introduced in 1907 by Vladimir Steklov (Stekloff) [13] by the equality


where h>0 is so small that the interval (x,x+h) lies in the domain of the definition of the locally integrable function f.

Theorem 9.

Let Lα,ρ be defined by (11). For every fCB(+) and x0, the following inequality




Let fCB(+) be arbitrarily fixed. For x=0, our relation is obvious. Let x>0. Applying the Steklov mean fh given at (19), we can write

|Lα,ρ(f;x)f(x)| Lα,ρ(|ffh|;x)+|Lα,ρ(fhfh(x);x)|
+|fh(x)f(x)|. (22)

Using the fact that the operators are non-expansive and taking in view (20), we obtain


Further, using successively Taylor’s expansion, the identity Lα,ρe0=e0 and relations (18), (21) we get

|Lα,ρ(fhfh(x);x)| fhμ2(Lα,ρ;x)+12fh′′μ2(Lα,ρ;x)

At this point we choose h:=2x(xu(x))>0 and returning at (3) we assemble the established increases. Our statement is fully motivated. ∎


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