In this paper we survey some recent results concerning the asymptotic behaviour of the iterates of a single Markov operator or of a sequence of Markov operators. Among other things, a characterization of the convergence of the iterates of Markov operators toward a given Markov projection is discussed in terms of the involved interpolation sets. Constructive approximation problems for strongly continuous semigroups of operators in terms of iterates are also discussed. In particular we present some simple criteria concerning their asymptotic behaviour. Finally, some applications are shown concerning Bernstein-Schnabl operators on convex compact sets and Bernstein-Durrmeyer operators with Jacobi weights on the unit hypercube. A final section contains some suggestions for possible further researches.
{"title":"Iterates of Markov Operators and Constructive Approximation of Semigroups","authors":"F. Altomare","doi":"10.33205/CMA.491601","DOIUrl":"https://doi.org/10.33205/CMA.491601","url":null,"abstract":"In this paper we survey some recent results concerning the asymptotic behaviour of the iterates of a single Markov operator or of a sequence of Markov operators. Among other things, a characterization of the convergence of the iterates of Markov operators toward a given Markov projection is discussed in terms of the involved interpolation sets. Constructive approximation problems for strongly continuous semigroups of operators in terms of iterates are also discussed. In particular we present some simple criteria concerning their asymptotic behaviour. Finally, some applications are shown concerning Bernstein-Schnabl operators on convex compact sets and Bernstein-Durrmeyer operators with Jacobi weights on the unit hypercube. A final section contains some suggestions for possible further researches.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49459859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In several recent papers, attempts have been made to apply Wardowski's method of $F$-contractions in order to obtain fixed point results for single and multivalued mappings in $b$-metric spaces. In this article, it is shown that in most cases the conditions imposed on respective mappings are too strong and that the results can be obtained directly, i.e., without using most of the properties of auxiliary function $F$.
{"title":"Notes on Some Recent Papers Concerning $F$-Contractions in $b$-Metric Spaces","authors":"Z. Kadelburg, S. Radenović","doi":"10.33205/cma.468813","DOIUrl":"https://doi.org/10.33205/cma.468813","url":null,"abstract":"In several recent papers, attempts have been made to apply Wardowski's method of $F$-contractions in order to obtain fixed point results for single and multivalued mappings in $b$-metric spaces. In this article, it is shown that in most cases the conditions imposed on respective mappings are too strong and that the results can be obtained directly, i.e., without using most of the properties of auxiliary function $F$.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42639020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper is a survey concerning representations for the remainder term of Bernstein-Schurer-Stancu and respectively Stancu (based on factorial powers) bivariate approximation formulas, using bivariate divided differences. As particular cases the remainder terms of bivariate Bernstein-Stancu, Schurer and classical Bernstein bivariate approximation formulas are obtained. Finally, one presents some mean value properties, similar to those of the remainder term of classical Bernstein univariate approximation formula.
{"title":"On the Remainder Term of Some Bivariate Approximation Formulas Based on Linear and Positive Operators","authors":"Dan Barbosu","doi":"10.33205/CMA.442151","DOIUrl":"https://doi.org/10.33205/CMA.442151","url":null,"abstract":"The paper is a survey concerning representations for the remainder term of Bernstein-Schurer-Stancu and respectively Stancu (based on factorial powers) bivariate approximation formulas, using bivariate divided differences. As particular cases the remainder terms of bivariate Bernstein-Stancu, Schurer and classical Bernstein bivariate approximation formulas are obtained. Finally, one presents some mean value properties, similar to those of the remainder term of classical Bernstein univariate approximation formula.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43610078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present paper, we define Jakimovski-Leviatan type modified operators. We study some approximation results for these operators. We also determine the order of convergence in terms of modulus of continuity, Lipschitz functions, Peetre's K-functional, second order modulus of continuity and weighted modulus of continuity.
{"title":"Approximation of Modified Jakimovski-Leviatan-Beta Type Operators","authors":"M. Mursaleen, M. Nasiruzzaman","doi":"10.33205/CMA.453284","DOIUrl":"https://doi.org/10.33205/CMA.453284","url":null,"abstract":"In the present paper, we define Jakimovski-Leviatan type modified operators. We study some approximation results for these operators. We also determine the order of convergence in terms of modulus of continuity, Lipschitz functions, Peetre's K-functional, second order modulus of continuity and weighted modulus of continuity.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44250069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We survey some results concerning differences of positive linear operators from Approximation Theory, and present some new results in this direction.
本文综述了逼近理论中关于正线性算子差分的一些结果，并在此方向上给出了一些新的结果。
{"title":"A Survey on Estimates for the Differences of Positive Linear Operators","authors":"A. Acu, Sever Hodiş, I. Raşa","doi":"10.33205/CMA.478408","DOIUrl":"https://doi.org/10.33205/CMA.478408","url":null,"abstract":"We survey some results concerning differences of positive linear operators from Approximation Theory, and present some new results in this direction.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69532114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present paper, we introduce the Bezier variant of the Srivastava-Gupta operators, which preserve constant as well as linear functions. Our study focuses on a direct approximation theorem in terms of the Ditzian-Totik modulus of smoothness, respectively the rate of convergence for differentiable functions whose derivatives are of bounded variation.
{"title":"On the Bézier Variant of the Srivastava-Gupta Operators","authors":"Arun Kajla","doi":"10.33205/CMA.465073","DOIUrl":"https://doi.org/10.33205/CMA.465073","url":null,"abstract":"In the present paper, we introduce the Bezier variant of the Srivastava-Gupta operators, which preserve constant as well as linear functions. Our study focuses on a direct approximation theorem in terms of the Ditzian-Totik modulus of smoothness, respectively the rate of convergence for differentiable functions whose derivatives are of bounded variation.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43398924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this short survey is to collect and combine basic notions and results in the fixed point theory in the context of $b$-metric spaces. It is also aimed to show that there are still enough rooms for several researchers in this interesting direction and a huge application potential.
{"title":"A Short Survey on the Recent Fixed Point Results on $b$-Metric Spaces","authors":"E. Karapınar","doi":"10.33205/CMA.453034","DOIUrl":"https://doi.org/10.33205/CMA.453034","url":null,"abstract":"The aim of this short survey is to collect and combine basic notions and results in the fixed point theory in the context of $b$-metric spaces. It is also aimed to show that there are still enough rooms for several researchers in this interesting direction and a huge application potential.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45112850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present work, our aim of this study is generalization and extension of the theory of interpolation of two dimensional functions to functionals or operators by means of Urysohn type nonlinear operators. In accordance with this purpose, we introduce and study a new type of Urysohn type nonlinear operators. In particular, we investigate the convergence problem for nonlinear operators that approximate the Urysohn type operator in two dimensional case. The starting point of this study is motivated by the important applications that approximation properties of certain families of nonlinear operators have in signal-image reconstruction and in other related fields. We construct our nonlinear operators by using a nonlinear form of the kernels together with the Urysohn type operator values instead of the sampling values of the function.
{"title":"Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators","authors":"H. Karsli","doi":"10.33205/CMA.453027","DOIUrl":"https://doi.org/10.33205/CMA.453027","url":null,"abstract":"In the present work, our aim of this study is generalization and extension of the theory of interpolation of two dimensional functions to functionals or operators by means of Urysohn type nonlinear operators. In accordance with this purpose, we introduce and study a new type of Urysohn type nonlinear operators. In particular, we investigate the convergence problem for nonlinear operators that approximate the Urysohn type operator in two dimensional case. The starting point of this study is motivated by the important applications that approximation properties of certain families of nonlinear operators have in signal-image reconstruction and in other related fields. We construct our nonlinear operators by using a nonlinear form of the kernels together with the Urysohn type operator values instead of the sampling values of the function.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44473461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present article, we study the approximation of difference of operators and find the quantitative estimates for the difference of Lupas operators with Lupas-Sz asz operators and Lupas-Kantorovich operators in terms of modulus of continuity. Also, we find the quantitative estimate for the difference of Lupas-Kantorovich operators and Lupas-Szasz operators.
{"title":"Differences of Operators of Lupaş Type","authors":"Vijay Gupta","doi":"10.33205/CMA.452962","DOIUrl":"https://doi.org/10.33205/CMA.452962","url":null,"abstract":"In the present article, we study the approximation of difference of operators and find the quantitative estimates for the difference of Lupas operators with Lupas-Sz asz operators and Lupas-Kantorovich operators in terms of modulus of continuity. Also, we find the quantitative estimate for the difference of Lupas-Kantorovich operators and Lupas-Szasz operators.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47457503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce Kantorovich type modification of $(p, q)$-Meyer-K o nig-Zeller operators. We estimate rate of convergence of proposed operators using modulus of continuity and Lipschitz class functions. Further, we obtain the statistical convergence and local approximation results for these operators. In the last section, we estimate the rate of convergence of $(p, q)$-Meyer-K o nig-Zeller Kantorovich operators by means of Matlab programming.
本文介绍了$（p，q）$-Meyer-K o nig-Zeller算子的Kantorovich型修改。我们使用连续模和Lipschitz类函数来估计所提出的算子的收敛速度。此外，我们还得到了这些算子的统计收敛性和局部逼近结果。在最后一节中，我们通过Matlab编程估计了$（p，q）$-Meyer-K o nig-Zeller-Kantorovich算子的收敛速度。
{"title":"Approximation Properties of Kantorovich Type Modifications of $(p, q)-$Meyer-König-Zeller Operators","authors":"Ramapati Maurya, Honey Sharma, Cheeena Gupta","doi":"10.33205/CMA.436071","DOIUrl":"https://doi.org/10.33205/CMA.436071","url":null,"abstract":"In this paper, we introduce Kantorovich type modification of $(p, q)$-Meyer-K o nig-Zeller operators. We estimate rate of convergence of proposed operators using modulus of continuity and Lipschitz class functions. Further, we obtain the statistical convergence and local approximation results for these operators. In the last section, we estimate the rate of convergence of $(p, q)$-Meyer-K o nig-Zeller Kantorovich operators by means of Matlab programming.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48521536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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