Differences of Positive Linear Operators on Simplices

Acu, Ana-Maria; Başcanbaz-Tunca, Gülen; Rasa, Ioan

doi:https://doi.org/10.1155/2021/5531577

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Approximation Methods: Theory and Applications

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Volume 2021 | Article ID 5531577 | https://doi.org/10.1155/2021/5531577

Differences of Positive Linear Operators on Simplices

Ana-Maria Acu,¹Gülen Başcanbaz-Tunca,²and Ioan Rasa³

Academic Editor: John R. Akeroyd

Received07 Jan 2021

Revised09 Mar 2021

Accepted13 Mar 2021

Published25 Mar 2021

Abstract

The aim of the paper is twofold: we introduce new positive linear operators acting on continuous functions defined on a simplex and then estimate differences involving them and/or other known operators. The estimates are given in terms of moduli of smoothness and -functionals. Several applications and examples illustrate the general results.

1. Introduction

Differences of positive linear operators were intensively investigated in the last years; see [1–14] and the references therein. The operators involved in these studies act usually on continuous functions defined on real intervals, and the differences are estimated in terms of moduli of smoothness and -functionals. In some papers, operators having equal central moments up to a certain order are considered. Other articles deal with operators constructed with the same fundamental functions and different functionals in front of them.

The study of differences of positive linear operators is important from a theoretical point of view, but also from a practical one. Let and be certain positive linear operators. If we know that is small, we can choose or taking into account other qualities of them like shape-preserving properties and smoothness/Lipschitz preserving properties.

This paper is concerned with differences of positive linear operators acting on continuous functions defined on simplices. For the sake of simplicity, we consider only the case of the canonical simplex in , where the notation is simpler, but the results can be easily translated to an arbitrary simplex in .

We consider the bivariate versions of some classical operators like Bernstein, Durrmeyer, Kantorovich, and genuine Bernstein-Durrmeyer operators. These bivariate versions were already studied in literature from other points of view. We introduce the bivariate versions of other operators: (see [15, 16]) and the operators defined in [17]. All these operators are constructed with the fundamental Bernstein polynomials on the two-dimensional simplex. A different kind of operator is the bivariate version of the univariate Beta operator of Mühlbach and Lupas (see [18–20]); we introduce it and use it in composition with the Bernstein operator to get a useful representation of .

We get estimates of differences of the abovementioned operators, in terms of suitable moduli of smoothness and -functionals.

To resume, the aim of our paper is twofold: we introduce new operators on a simplex and then estimate differences involving them and other known operators.

The list of applications and examples can be enlarged. In particular, we will be interested for a future work in studying differences of bivariate versions of operators, which preserve exponential functions (see [21–23]). We also intend to deepen the study of the newly introduced Beta operators on the simplex and to consider the composition of it with other operators, leading to new applications and-why not-new theoretical aspects/problems. Given a Markov operator (i.e., a positive linear operator which preserves the constant functions), the study of its iterate is important not only in Approximation Theory but also in Ergodic Theory and other areas of research. We intend to investigate from this point of view the newly introduced operators, which are in fact Markov operators.

We end this Introduction by presenting some notation and a fundamental inequality expressed in Lemma 1. Section 2 contains the main theoretical results, while Section 3 is devoted to applications and examples.

Let be the canonical simplex in and denote a space of real-valued continuous functions of two variables defined on , containing the polynomials. Throughout the paper, we will denote by the constant function, namely, and ,will denote the th coordinate functions restricted on , which are given by

Let be a positive linear functional such that . Set

Then, one has

Let be the space of all real-valued (continuous) functions, differentiable on and whose partial derivatives of order can be continuously extended to , having

Lemma 1. If , then

Proof. Consider the line segment connecting with . From Taylor’s formula (see [24], p.245), there is a point on this line segment, different from and , such that Therefore, we can write which gives the result.

2. Difference of Bivariate Positive Linear Operators

Denote by the space of real-valued continuous functions on with the norm . Let be a set of nonnegative integers and for let satisfy . Let and ,, be positive linear functionals such that and . Moreover, let be the set of all for which

Now, consider the bivariate positive linear operators and acting from into defined, for ,by respectively. For future correspondences, we denote where is the -norm in .

In the following, we adopt the definitions of -functional and modulus of smoothness from [25, 26]. Let

For , th order differences on the subset are defined as

The th order modulus of smoothness of is a function given by

Let be the space of all real-valued (continuous) functions, differentiable on and whose partial derivatives of order can be continuously extended to , with the seminorm

For , we shall use the following -functional:

Then, there exist such that for any (see [25, 26])

Here, depends only on (for the general definition onthe spaces of functions on bounded domains, see [25] or, on unbounded domains see [27], p.341.

Theorem 2. If , then where is defined in Lemma 1.

Proof. Let . From Lemma 1, we get

Theorem 3. If , then where , and

Proof. Let . From Theorem 2, we get where is the same notation as in Lemma 1 for . Since partial derivatives of exist and are continuous everywhere in , it follows that is differentiable at every point of the line segment connecting the points and in , . By the mean value theorem (see, e.g., [24], p. 239), there is a point on this line segment such that From (16), we get Moreover, since , (22) gives that Finally, from (18), we obtain

3. Applications

3.1. Difference of Bivariate Bernstein Operators and Their Durrmeyer Variants

For every , and , the th bivariate Bernstein operator is defined by where with , (see, e.g., [28], p. 115).

For , the bivariate Durrmeyer operators are defined by see, e.g., [29].

Now, denoting the bivariate Bernstein operators and bivariate Durrmeyer operators can be written as respectively.

Proposition 4. For bivariate Bernstein operators and their Durrmeyer variants, the following properties hold: (i)If , thenwhere is the same as in Lemma 1 and (ii)If , then

Proof. We need to evaluate the terms in (11). So, we get the following results: . Therefore, we easily obtain that Using Maple, one obtains It is easy to verify that .
Now, for , we obtain The rest of the proof follows from Theorems 2 and 3.

3.2. Difference of Bivariate Bernstein Operators and the Bivariate Operators

Let be the space of polynomials over of degree at most . In [17], Aldaz et al. introduced a Bernstein operator that fixes and . The operators are given by

Here, for and , we introduce the bivariate form of the operators as follows

Denoting for , , we get

Proposition 5. For bivariate Bernstein operators and bivariate operators , the following properties hold: (i)If , then(ii)If , then

3.3. Difference of Bivariate Bernstein Operators and Bivariate Genuine Bernstein-Durrmeyer Operators

In 1987, Chen [30] and Goodman and Sharma [31] constructed the following positive linear operators where , and

For the historical background of these operators, we refer to [32]. In 1991, Goodman and Sharma [33] constructed and studied the multivariate form of the operators on a simplex. In [34], Sauer deeply studied the multivariate genuine Bernstein-Durrmeyer operators. Here, for , we consider the bivariate form given by with the bivariate Bernstein’s fundamental functions given by (28) (see [33], Formula 1.7). These operators satisfy at the vertices of .

Proposition 6. For bivariate Bernstein operators and bivariate genuine Bernstein-Durrmeyer operators, the following properties hold: (i)If , thenwhere is the same as in Lemma 1 and (ii)If , then

Proof. If we denote then for the bivariate Bernstein operators, we have The bivariate genuine Bernstein-Durrmeyer operators are given by Now, for , we get Hence, we obtain Therefore, and The proof is concluded by using Theorems 2 and 3.

4. The Difference

Let and . The operators are introduced by Paltanea in [35] (see also [15, 16]). These operators are defined by where , , and are Euler’s Beta function.

Here, for , we consider the bivariate form of these operators, given by where

It can be easily seen that, for , we obtain the genuine Bernstein-Durrmeyer operators . On the other hand, these operators have the following limiting behavior.

Theorem 7. For any , one has

Proof. Let , . Then, Since we get Similar results can be obtained for .
Using Korovkin’s theorem (see [36], p. 534, C.4.3.3), it follows . Therefore,

Proposition 8. For the bivariate operators , the following properties hold: (i)If , thenwhere is the same as in Lemma 1 and (ii)If , then

Proof. Since , , we get Therefore, and

5. Difference of Bivariate Bernstein Operators and Their Kantorovich Variants

In 2017, F. Altomare et al. [37] introduced Kantorovich operators on as follows where is given by (28). It can be easily seen that, for , we obtain Kantorovich operators introduced in [38].

If we denote the bivariate Kantorovich operators can be written as

Proposition 9. For bivariate Bernstein operators and bivariate Bernstein-Kantorovich operators, the following properties hold: (i)If , thenwhere is the same as in Lemma 1 and (ii)If , then

Proof. As in the previous examples, taking Bernstein operators as we get Therefore, we easily obtain that Then Moreover, we have Then, the proof follows from Theorems 2 and 3.

6. A Beta Operator on

For , , and , let us define

For , this is the bivariate version of the operator ; see [39] and the references therein.

Theorem 10. is a positive linear operator acting between and . Moreover, and if , , , , integers, then

Proof. It is easy to prove (81) and (82) by direct calculation. It remains to prove that if , then, . To do this, it suffices to verify that is continuous at each point of the boundary of . Let us prove that if then Let , . For define , , . Then, and are positive linear functionals of norm .
Let . Then, there exists a polynomial function on such that . Using (82), it is easy to verify that Consequently, there exists with So, if , we have This shows that and then (83) is proved.
The continuity of at the other boundary points can be proved similarly.

Proposition 11. For each , one has

Proof. Using Theorem 10, it is easy to verify that (88) is valid for the functions . But these functions form a Korovkin test system (see [36], p. 534, C.4.3.3), so that (88) holds for each .

In what follows, we formulate a

Conjecture 12. If is convex and , then, the function is decreasing on .
It is supported by the following facts. (i)The unidimensional version of the conjecture is valid: see [14, 40].(ii) is a positive linear operator preserving the affine functions; this impliesNow, (88) combined with (89) support the conjecture. (iii)The conjecture is valid for the functions In the sequel, we present two results under the hypothesis that the conjecture is true. To this end, let us introduce some notation.
Let and Then, the functions and are convex on ; indeed, for each of them, the Hessian matrix is positive semidefinite.

Theorem 13. If and , then

Proof. Let . If the Conjecture is true, we have and . Thus Consequently, Moreover, Now, So, combining (93) and (95), we have proved the theorem.

Theorem 14. If and , then

Proof. It is easy to verify that and, if , then Using these facts and supposing that the conjecture is true, we have Now, Thus, From , we get a similar upper bound for , which concludes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no competing financial interests.

Acknowledgments

This work was supported by a Hasso Plattner Excellence Research Grant (LBUS-HPI-ERG-2020-04), financed by the Knowledge Transfer Center of the Lucian Blaga University of Sibiu.

References

A. M. Acu and I. Rasa, “New estimates for the differences of positive linear operators,” Numerical Algorithms, vol. 73, no. 3, pp. 775–789, 2016.
View at: Publisher Site | Google Scholar
A. M. Acu, S. Hodis, and I. Rasa, “A survey on estimates for the differences of positive linear operators,” Constructive Mathematical Analysis, vol. 1, no. 2, pp. 113–127, 2018.
View at: Google Scholar
A. M. Acu and I. Rasa, “Estimates for the differences of positive linear operators and their derivatives,” Numerical Algorithms, vol. 85, no. 1, pp. 191–208, 2020.
View at: Publisher Site | Google Scholar
A. Aral, D. Inoan, and I. Rasa, “On differences of linear positive operators,” Analysis and Mathematical Physics, vol. 9, no. 3, pp. 1227–1239, 2019.
View at: Publisher Site | Google Scholar
H. Gonska, P. Pitul, and I. Rasa, “On Peano's form of the Taylor remainder, Voronovskaja's theorem and the commutator of positive linear operators,” in Numerical Analysis and Approximation Theory (Proc. Int. Conf. Cluj-Napoca 2006), O. Agratini and P. Blaga, Eds., pp. 55–80, Cluj-Napoca, Casa Cartii de Stiintă, 2006.
View at: Google Scholar
H. Gonska, P. Piƫul, and I. Raşa, “On differences of positive linear operators,” Carpathian Journal of Mathematics, vol. 22, no. 1-2, pp. 65–78, 2006.
View at: Google Scholar
H. Gonska and I. Raşa, “Differences of positive linear operators and the second order modulus,” Carpathian Journal of Mathematics, vol. 24, no. 3, pp. 332–340, 2008.
View at: Google Scholar
V. Gupta, D. Agrawal, and M. T. Rassias, “Quantitative estimates for differences of Baskakov-Type operators,” Complex Analysis and Operator Theory, vol. 13, no. 8, pp. 4045–4064, 2019.
View at: Publisher Site | Google Scholar
V. Gupta, “Estimate for the difference of operators having different basis functions,” Rendiconti del Circolo Matematico di Palermo Series 2, vol. 69, no. 3, pp. 995–1003, 2020.
View at: Publisher Site | Google Scholar
V. Gupta, A. M. Acu, and H. M. Srivastava, “Difference of some positive linear approximation operators for higher-order derivatives,” Symmetry, vol. 12, no. 6, p. 915, 2020.
View at: Publisher Site | Google Scholar
V. Gupta, “On difference of operators with applications to Szász type operators,” RACSAM, vol. 113, no. 3, pp. 2059–2071, 2019.
View at: Publisher Site | Google Scholar
V. Gupta and G. Tachev, “A note on the differences of two positive linear operators,” Constructive Mathematical Analysis, vol. 2, no. 1, pp. 1–7, 2019.
View at: Publisher Site | Google Scholar
V. Gupta, T. M. Rassias, P. N. Agrawal, and A. M. Acu, “Estimates for the differences of positive linear operators,” in Recent Advances in Constructive Approximation Theory. Springer Optimization and Its Applications, vol 138, Springer, Cham, 2018.
View at: Publisher Site | Google Scholar
I. Rasa and E. D. Stanila, “On some operators linking the Bernstein and the genuine Bernstein Durrmeyer operators,” Journal of Applied Functional Analysis, vol. 9, no. 3-4, pp. 369–378, 2014.
View at: Google Scholar
H. Gonska and R. Paltanea, “Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions,” Czechoslovak Mathematical Journal, vol. 60, no. 3, pp. 783–799, 2010.
View at: Publisher Site | Google Scholar
H. Gonska and R. Paltanea, “Quantitative convergence theorems for a class of Bernstein-Durrmeyer operators preserving linear functions,” Ukrainian Mathematical Journal, vol. 62, pp. 913–922, 2010.
View at: Google Scholar
J. M. Aldaz, O. Kounchev, and H. Render, “Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces,” Numerische Mathematik, vol. 114, no. 1, pp. 1–25, 2009.
View at: Publisher Site | Google Scholar
A. Lupas, Die Folge der Betaoperatoren, Dissertation, Universität Stuttgart, 1972.
G. Mühlbach, “Verallgemeinerungen der Bernstein- und der Lagrangepolynome, Bemerkungen zu einer Klasse linearer Polynomoperatoren von D. D. Stancu,” Revue Roumaine des Mathematiques Pures et Appliquees, vol. 15, pp. 1235–1252, 1970.
View at: Google Scholar
G. Mühlbach, “Rekursionsformeln für die zentralen Momente derPólya- und der Beta-Verteilung,” Metrika, vol. 19, no. 1, pp. 171–177, 1972.
View at: Publisher Site | Google Scholar
T. Acar, A. Aral, and H. Gonska, “On Szász–Mirakyan operators preserving e^2ax, a > 0,” Mediterranean Journal of Mathematics, vol. 14, no. 1, p. 6, 2017.
View at: Publisher Site | Google Scholar
A. M. Acu, A. Aral, and I. Rasa, “Generalized Bernstein Kantorovich operators: Voronovskaya type results, convergence in variation,” Carpathian J. Math, vol. 37, no. 3, 2021.
View at: Google Scholar
A. M. Acu, A. I. Maduta, and I. Rasa, “Voronovskaya type results and operators fixing two functions,” Mathematical Modelling and Analysis, (in press).
View at: Google Scholar
R. G. Bartle, The Elements of Real Analysis, 1964, John Wiley & Sons Ltd.
S. Dekel, D. Leviatan, and M. Sharir, “On bivariate smoothness spaces associated with nonlinear approximation,” Constructive Approximation, vol. 20, no. 4, pp. 625–646, 2004.
View at: Publisher Site | Google Scholar
H. Johnen and K. Scherer, “On the equivalence of the K-functional and the moduli of continuity and some applications,” in Constructive Theory of Functions of Several Variables. Lecture Notes in Math, vol 571, pp. 119–140, Springer-Verlag, Berlin, 1976.
View at: Publisher Site | Google Scholar
C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988.
F. Altomare, “Korovkin-type theorems and approximation by positive linear operators,” Surveys in Approximation Theory, vol. 5, pp. 92–164, 2010.
View at: Google Scholar
M.-M. Derriennic, “On multivariate approximation by Bernstein-type polynomials,” Journal of Approximation Theory, vol. 45, no. 2, pp. 155–166, 1985.
View at: Publisher Site | Google Scholar
W. Chen, “On the modified Durrmeyer–Bernstein operator,” in Report of the Fifth Chinese Conference on Approximation Theory (in Chinese, Handwritten), Zhen Zhou, China, 1987.
View at: Google Scholar
T. N. T. Goodman and A. Sharma, “A modified Bernstein-Schoenberg operator,” in Proceedings of the Conference on Constructive Theory of Functions, pp. 166–173, Publ. House Bulg. Acad. of Sci., Sofia, 1988.
View at: Google Scholar
H. Gonska, D. Kacsó, and I. Rasa, “The genuine Bernstein–Durrmeyer operators revisited,” Results in Mathematics, vol. 62, no. 3-4, pp. 295–310, 2012.
View at: Publisher Site | Google Scholar
T. N. T. Goodman and A. Sharma, “A Bernstein type operator on the simplex,” Mathematica Balkanica, vol. 5, pp. 129–145, 1991.
View at: Google Scholar
T. Sauer, “The genuine Bernstein-Durrmeyer operator on a simplex,” Results in Mathematics, vol. 26, no. 1-2, pp. 99–130, 1994.
View at: Publisher Site | Google Scholar
R. Paltanea, “A class of Durrmeyer type operators preserving linear functions,” Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity, vol. 5, pp. 109–117, 2007.
View at: Google Scholar
F. Altomare and M. Campiti, “Korovkin-type approximation theory and its applications,” Tech. Rep., Series: De Gruyter Studies in Mathematics, 1994.
View at: Google Scholar
F. Altomare, M. Cappelletti Montano, V. Leonessa, and I. Rasa, “A generalization of Kantorovich operators for convex compact subsets,” Banach Journal of Mathematical Analysis, vol. 11, no. 3, pp. 591–614, 2017.
View at: Publisher Site | Google Scholar
D.-X. Zhou, “Converse theorems for multidimensional Kantorovich operators,” Analysis Mathematica, vol. 19, no. 1, pp. 85–100, 1993.
View at: Publisher Site | Google Scholar
H. Gonska, D. Kacsó, and I. Rasa, “On the composition and decomposition of positive linear operators (VI),” in Constructive Theory of Functions, Sozopol 2016, K. Ivanov, G. Nikolov, and R. Uluchev, Eds., pp. 1–15, Prof. Marin Drinov Academic Publishing House, Sofia, 2018.
View at: Google Scholar
J. A. Adell, F. G. Badia, J. de la Cal, and F. Plo, “On the property of monotonic convergence for beta operators,” Journal of Approximation Theory, vol. 84, no. 1, pp. 61–73, 1996.
View at: Publisher Site | Google Scholar

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Copyright © 2021 Ana-Maria Acu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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