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

Octavian Agratini
(original), Approximation Theory, html, JCR/ISI, paper, PDF, works

Abstract

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.

 

Authors

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

Keywords

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

PDF

https://ictp.acad.ro/wp-content/uploads/2023/03/2021-Abel-Agratini-On-the-Durrmeyer-type-variant-and-generalization-of-Lototsky-Bernstein-operators-symmetry.pdf

also freely available at the publisher: https://doi.org/10.3390/sym13101841

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

About this paper

Journal

Symmetry

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MDPI

DOI

https://doi.org/10.3390/sym13101841

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

Octavian Agratini  and  Ulrich Abel
Abstract

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.

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

1

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

(Bn⁢f)⁢(x)=∑k=0nbn,k⁢(x)⁢f⁢(kn),x∈[0,1],

where bn,k⁢(x)=\binom⁢n⁢k⁢xk⁢(1−x)n−k, 0≤k≤n, 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]→ℝ 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∈ℕ, let hj:[0,1]→[0,1] be a continuous function. Further, for each n∈ℕ is defined a system of functions (an,k)k=0,n¯ for x∈[0,1] by the relation

∏j=1n(hj⁢(x)⁢y+1−hj⁢(x))=∑k=0nan,k⁢(x)⁢yk,y∈ℝ. (1)

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

an,k⁢(x)=∑\substack⁢K∪K¯=ℕn⁢C⁢a⁢r⁢d⁢(K)=k∏i∈K¯(1−hi⁢(x))⁢∏j∈Khj⁢(x), (2)

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

(Ln⁢f)⁢(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∈ℕ, they are also positive. It is clear that in the special case hj⁢(x)=x, j∈ℕ, the functions (an,k)0≤k≤n, n∈ℕ, become Bernstein bases (pn,k)0≤k≤n, n∈ℕ, consequently Ln operator turns into Bn operator.

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

limn→∞1n⁢∑i=1nhi⁢(x)=x⁢ uniformly with respect to ⁢x⁢ on ⁢[0,1], (4)

then

limn→∞(Ln⁢f)⁢(x)=f⁢(x)⁢ uniformly with respect to ⁢x⁢ on ⁢[0,1] (5)

for every f∈C⁢([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∈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 ℕ0={0}∪ℕ and ej, j∈ℕ0, monomials of degree j, e0⁢(x)=1, ej⁢(x)=xj, j∈ℕ. The statement (5) is motivated by Bohman-Korovkin criterion which says: If a sequence of linear positive operators (Λn)n≥1 defined on C⁢([a,b]) has the property that (Λn⁢ek)n≥1 converges to ek uniformly on [a,b], k∈{0,1,2}, then (Λn⁢f)n≥1 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

(Ln⁢e0)⁢(x)=Pn⁢(x;1)=1, (6)
(Ln⁢e1)⁢(x)=1n⁢∂Pn⁢(x;1)∂y=1n⁢∑i=1nhi⁢(x), (7)
(Ln⁢e2)⁢(x) =1n2⁢(∂2Pn⁢(x;1)∂y2+∂Pn⁢(x;1)∂y)
=(1n⁢∑i=1nhi⁢(x))2+1n2⁢∑i=1nhi⁢(x)⁢(1−hi⁢(x)). (8)

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

Moreover, for any k≥1,

(Ln⁢ek)⁢(x)=1nk⁢∑j=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 1≤j≤k, its closed form is given as follows

s⁢(k,j)=1j!⁢∑ν=0j(−1)j−ν⁢\binom⁢j⁢ν⁢νk,

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∈ℕ. 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, ‖f‖1=∫01|f⁢(x)|⁢𝑑x. Define Dn:L1⁢([0,1])→C⁢([0,1]) by formula

(Dn⁢f)⁢(x)=(n+1)⁢∑k=0nan,k⁢(x)⁢∫01pn,k⁢(t)⁢f⁢(t)⁢𝑑t,x∈[0,1]. (10)
{Remark}

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

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

(Dn⁢f)⁢(x)=∫01kn∗⁢(x,t)⁢f⁢(t)⁢𝑑t,x∈[0,1],

where

kn∗⁢(x,t)=(n+1)⁢∑k=0nan,k⁢(x)⁢pn,k⁢(t),(x,t)∈[0,1]×[0,1].

(c) If f∈C⁢([0,1]), then ‖Dn⁢f‖≤‖f‖, n∈ℕ, consequently the operators are non expansive in the space C⁢([0,1]).

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

{Lemma}

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

Dn⁢e0=e0, (11)
Dn⁢e1=nn+2⁢Ln⁢e1+1n+2, (12)
Dn⁢e2=n2(n+2)⁢(n+3)⁢Ln⁢e2+3⁢n(n+2)⁢(n+3)⁢Ln⁢e1+2(n+2)⁢(n+3) (13)

take place for each n∈ℕ.

Proof 2.1.

Using Beta function, for any p∈ℕ0 we deduce

∫01pn,k⁢(t)⁢tp⁢𝑑t=(k+p)!k!⁢n!(n+p+1)!,k=0,n¯.

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∈ℕ, i.e. Dn⁢φxj, where

φx⁢(t)=t−x,(t,x)∈[0,1]×[0,1].
{Lemma}

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

References

[1] King, J.P., The Lototsky transform and Bernstein polynomials. Canad. J. Math. 1966, 18, 89–91. [Google Scholar] [CrossRef]
[2] Xu, X.-W.; Goldman, R., On Lototsky-Bernstein operators and Lototsky-Bernstein bases. Comput. Aided Geom. Design 2019, 68, 48–59. [Google Scholar] [CrossRef]
[3] Goldman, R.; Xu, X.-W.; Zeng, X.-M., Applications of the Shorgin identity to Bernstein type operators. Results Math. 2018, 73, 2. [Google Scholar] [CrossRef]
[4] Xu, X.-W.; Zeng, X.-M.; Goldman, R., Shape preserving properties of univariate Lototsky-Bernstein operators. J. Approx. Theory 2017, 224, 13–42. [Google Scholar] [CrossRef]
[5] Popa, D., Intermediate Voronovskaja type results for the Lototsky-Bernstein type operators. RACSAM 2020, 114, 12. [Google Scholar] [CrossRef]
[6] Eisenberg, S.; Wood, B., Approximation of analytic functions by Bernstein-type operators. J. Approx. Theory 1972, 6, 242–248. [Google Scholar] [CrossRef]
[7] Abramowitz, M.; Stegun, I.A., (Eds.) Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables; Series 55; National Bureau of Standards Applied Mathematics: Washington, DC, USA, 1964. Available online: https://eric.ed.gov/?id=ED250164 (accessed on 30 July 2021).
[8] Derriennic, M.M., Sur l’approximation de fonctions intégrables sur [0,1] par des polynômes de Bernstein modifies. J. Approx. Theory 1981, 31, 325–343. [Google Scholar] [CrossRef]
[9] Shisha, O.; Mond, B., The degree of convergence of linear positive operators. Proc. Natl. Acad. Sci. USA 1968, 60, 1196–1200. [Google Scholar] [CrossRef] [PubMed]
[10] DeVore, R.A.; Lorentz, G.G., Constructive Approximation; Grundlehren der Mathematischen Wissenschaften; Springer: Berlin, Germany, 1993; Volume 303. [Google Scholar]
[11] Mitjagin, B.S.; Semenov, E.M., Lack of interpolation of linear operators in spaces of smooth functions. Math. USSR. Izv. 1977, 11, 1229–1266. [Google Scholar] [CrossRef]
[12] Johnen, H., Inequalities connected with the moduli of smoothness. Mat. Vesnik 1972, 9, 289–305. [Google Scholar]
[13] Altomare, F., Korovkin-type theorems and approximation by positive linear operators. Surv. Approx. Theory 2010, 5, 92–164. [Google Scholar]
[14] Šaškin, Y.A., Korovkin systems in spaces of continuous functions. Izv. Akad. Nauk SSSR Ser. Mat. 1962, 26, 495–512. translated in Amer. Math. Soc. Transl. 1966, 54, 125–144. (In Russian) [Google Scholar]
[15] Censor, E., Quantitative results for positive linear approximation operators. J. Approx. Theory 1971, 4, 442–450. [Google Scholar] [CrossRef]
[16] Deo, N.; Kumar, S., Durrmeyer variant of Apostol-Genocchi-Baskakov operators. Quaes. Math. 2020. [Google Scholar] [CrossRef]
[17] Garg, T.; Ispir, N.; Agrawal, P.N., Bivariate q-Bernstein-Chlodowsky-Durrmeyer type operators and associated GBS operators. Asian-Eur. J. Math. 2020, 13, 2050091. [Google Scholar] [CrossRef]
[18] Kajla, A.; Mursaleen, M.; Acar, T., Durrmeyer type generalization of parametric Bernstein operators. Symmetry 2020, 12, 1141. [Google Scholar] [CrossRef]
[19] Neer, T.; Acu, A.M.; Agrawal, P.N., Baskakov-Durrmeyer type operators involving generalized Appell polynomials. Math. Methods Appl. Sci. 2020, 43, 2911–2923. [Google Scholar] [CrossRef]
[20] Abel, U.; Leviatan, D.; Rasa, I., On the q-monotonicity preservation of Durrmeyer-type operators. Mediterr. J. Math. 2021, 18, 173. [Google Scholar] [CrossRef]
[21] Agratini, O.; Aral, A.,  Approximation of some classes of functions by Landau type operators. Results Math. 2021, 76, 12. [Google Scholar] [CrossRef]

2021

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