On the Durrmeyer-type variant and generalizations of Lototsky–Bernstein operators


The starting points of the paper are the classic Lototsky–Bernstein operators. We present an integral Durrmeyer-type extension and investigate some approximation properties of this new class. The evaluation of the convergence speed is performed both with moduli of smoothness and with K-functionals of the Peetre-type. In a distinct section we indicate a generalization of these operators that is useful in approximating vector functions with real values defined on the hypercube. The study involves achieving a parallelism between different classes of linear and positive operators, which will highlight a symmetry between these approximation processes.



Ulrich Abel
Fachbereich MND, Technische Hochschule Mittelhessen, Germany

Octavian Agratini
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy


Lototsky operator; Korovkin theorem; modulus of smoothness; K-functional; Durrmeyer extension


Cite this paper as:

U. Abel, O. Agratini, On the Durrmeyer-type variant and generalizations of Lototsky–Bernstein operators, Symmetry 2021, 13 (10),  https://doi.org/10.3390/sym13101841

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1 Introduction

On the Durrmeyer-Type Variant and Generalizations of Lototsky–Bernstein Operators

Octavian Agratini  and  Ulrich Abel

The starting point of the paper is the classic Lototsky-Bernstein operators. We present an integral Durrmeyer type extension and investigate some approximation properties of this new class. The evaluation of the convergence speed is performed both with moduli of smoothness and with K-functionals of the Peetre’s type. In a distinct section we indicate a generalization of these operators useful in approximating vector functions with real values defined on the hypercube [0,1]q, q>1.

Lototsky operator; Korovkin theorem; modulus of smoothness; K-functional; Durremeyer extension


1 Introduction

It is widely acknowledged that the most studied linear positive operators are Bernstein operators which have known innumerable generalization over time.

Bernstein operators Bn:C([0,1])C([0,1]) are defined by


where bn,k(x)=\binomnkxk(1x)nk, 0kn, represent the n+1 Bernstein basis polynomials of degree n. As usual, C([0,1]) denotes the real Banach space of all continuous functions f:[0,1] endowed with the sup-norm , f=supx[0,1]|f(x)|. With the same norm we endow B([0,1]), the space of bounded real valued functions defined on [0,1].

Based on the generalized Lototsky matrix, an extension of these operators was given by King [1]. We present it in the following. For each j, let hj:[0,1][0,1] be a continuous function. Further, for each n is defined a system of functions (an,k)k=0,n¯ for x[0,1] by the relation

j=1n(hj(x)y+1hj(x))=k=0nan,k(x)yk,y. (1)

From the above identity we immediately obtain the coefficient of yk, k=0,n¯, that is

an,k(x)=\substackKK¯=nCard(K)=kiK¯(1hi(x))jKhj(x), (2)

where n={1,2,,n} and K¯=nK. For each real valued function f defined on [0,1], the n-th Lototsky-Bernstein operator is defined as follows

(Lnf)(x)=k=0nan,k(x)f(kn), (3)

see [1, Eq. 4]. Ln operators are linear. Since hj([0,1])[0,1] for all j, they are also positive. It is clear that in the special case hj(x)=x, j, the functions (an,k)0kn, n, become Bernstein bases (pn,k)0kn, n, consequently Ln operator turns into Bn operator.

King has established the sufficient condition on (hj)j sequence to ensure that (Ln)n1 is an approximation process on C([0,1]), his result can be written as follows. If

limn1ni=1nhi(x)=x uniformly with respect to x on [0,1], (4)


limn(Lnf)(x)=f(x) uniformly with respect to x on [0,1] (5)

for every fC([0,1]).

In recent years the study of these operators has been deepened, see, for example, the papers of Ron Goldman, Xiao-Wei Xu, Xiao-Ming Zeng [2], [3].

The purpose of this note is to define and to establish approximation properties for a Durmmeyer-type extension of Ln, n, operators. We mention that, using elements of probability theory, a Kantorovich-type extension was achieved in 2020 by Popa [4]. A second goal of this note is to extend the univariate operators for vector functions with real values.

2 Dn Operators

Set 0={0} and ej, j0, monomials of degree j, e0(x)=1, ej(x)=xj, j. The statement (5) is motivated by Bohman-Korovkin criterion which says: If a sequence of linear positive operators (Λn)n1 defined on C([a,b]) has the property that (Λnek)n1 converges to ek uniformly on [a,b], k{0,1,2}, then (Λnf)n1 converges to f uniformly on [a,b] for each f belonging to C([a,b]). In relation (1) let us denote by Pn(x;y) the polynomial of degree n in y and with the parameter x[0,1]. Following [5] we get

(Lne0)(x)=Pn(x;1)=1, (6)
(Lne1)(x)=1nPn(x;1)y=1ni=1nhi(x), (7)
(Lne2)(x) =1n2(2Pn(x;1)y2+Pn(x;1)y)
=(1ni=1nhi(x))2+1n2i=1nhi(x)(1hi(x)). (8)

Taking in view the mentioned criterion, the proof of (5) is completed.

Moreover, for any k1,

(Lnek)(x)=1nkj=1ks(k,j)jPn(x;1)yj, (9)

see [5, Eq. (2.6)]. In the above s(k,j) denotes a Stirling number of the second kind. For 1jk, its closed form is given as follows


see, e.g., [6, p. 824]. For k=1 and k=2, from (9) we reobtain the identities (7) and (2).

Usually, in the papers that approached Lototsky operators, in order to obtain significant results, the authors imposed additional conditions on the functions hj, j. For similar reason, we define a particular Durrmeyer type construction that involves both Lototsky-Bernstein and genuine Bernstein bases. L1([0,1]) stands for the Banach space of all real valued integrable functions on [0,1] endowed with the norm 1, f1=01|f(x)|𝑑x. Define Dn:L1([0,1])C([0,1]) by formula

(Dnf)(x)=(n+1)k=0nan,k(x)01pn,k(t)f(t)𝑑t,x[0,1]. (10)

(a) The operators keep the properties of linearity and positivity.

(b) By using a bivariate kernel we can write Dnf in a more compact form as follows




(c) If fC([0,1]), then Dnff, n, consequently the operators are non expansive in the space C([0,1]).

(d) For the particular case hj=e1, j, Dn operators turn into the genuine Durrmeyer operators [7].


Let Dn, n, be the operators defined by (10). The following identities

Dne0=e0, (11)
Dne1=nn+2Lne1+1n+2, (12)
Dne2=n2(n+2)(n+3)Lne2+3n(n+2)(n+3)Lne1+2(n+2)(n+3) (13)

take place for each n.

Proof 2.1.

Using Beta function, for any p0 we deduce


By a straightforward calculation, the definition of Dn operators as well as the relation (3) lead us to the enunciated identities.

At this point we introduce the j-th central moment of Dn operators, j, i.e. Dnφxj, where


The second order central moments of the Dn, n, operators satisfy the following relations


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