In this note we establish a quantitative Voronovskaja theorem for modified Bernstein polynomials using the first order Ditzian-Totik modulus of smoothness.
{"title":"A Quantitative Variant of Voronovskaja's Theorem for King-Type Operators","authors":"Z. Finta","doi":"10.33205/CMA.553427","DOIUrl":"https://doi.org/10.33205/CMA.553427","url":null,"abstract":"In this note we establish a quantitative Voronovskaja theorem for modified Bernstein polynomials using the first order Ditzian-Totik modulus of smoothness.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49541476","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 and study a new sequence of positive linear operators, acting on both spaces of continuous functions as well as spaces of integrable functions on $[0, 1]$. We state some qualitative properties of this sequence and we prove that it is an approximation process both in $C([0, 1])$ and in $L^p([0, 1])$, also providing some estimates of the rate of convergence. Moreover, we determine an asymptotic formula and, as an application, we prove that certain iterates of the operators converge, both in $C([0, 1])$ and, in some cases, in $L^p([0, 1])$, to a limit semigroup. Finally, we show that our operators, under suitable hypotheses, perform better than other existing ones in the literature.
{"title":"A Sequence of Kantorovich-Type Operators on Mobile Intervals","authors":"M. C. Montano, V. Leonessa","doi":"10.33205/CMA.571078","DOIUrl":"https://doi.org/10.33205/CMA.571078","url":null,"abstract":"In this paper, we introduce and study a new sequence of positive linear operators, acting on both spaces of continuous functions as well as spaces of integrable functions on $[0, 1]$. We state some qualitative properties of this sequence and we prove that it is an approximation process both in $C([0, 1])$ and in $L^p([0, 1])$, also providing some estimates of the rate of convergence. Moreover, we determine an asymptotic formula and, as an application, we prove that certain iterates of the operators converge, both in $C([0, 1])$ and, in some cases, in $L^p([0, 1])$, to a limit semigroup. Finally, we show that our operators, under suitable hypotheses, perform better than other existing ones in the literature.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42184046","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}
Some inequalities for synchronous functions that are a mixture between Cebysev’s and Jensen's inequality are provided. Applications for $f$ -divergence measure and some particular instances including Kullback-Leibler divergence, Jeffreys divergence and $chi ^{2}$-divergence are also given.
{"title":"Inequalities for Synchronous Functions and Applications","authors":"S. Dragomir","doi":"10.33205/CMA.562166","DOIUrl":"https://doi.org/10.33205/CMA.562166","url":null,"abstract":"Some inequalities for synchronous functions that are a mixture between Cebysev’s and Jensen's inequality are provided. Applications for $f$ -divergence measure and some particular instances including Kullback-Leibler divergence, Jeffreys divergence and $chi ^{2}$-divergence are also given.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48544248","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 paper we introduce a general class of linear positive approximation processes defined on bounded and unbounded intervals designed using an appropriate function. Voronovskaya type theorems are given for these new constructions. Some examples including well known operators are presented.
{"title":"Positive Linear Operators Preserving $tau $ and $tau ^{2}$","authors":"T. Acar, A. Aral, I. Raşa","doi":"10.33205/CMA.547221","DOIUrl":"https://doi.org/10.33205/CMA.547221","url":null,"abstract":"In the paper we introduce a general class of linear positive approximation processes defined on bounded and unbounded intervals designed using an appropriate function. Voronovskaya type theorems are given for these new constructions. Some examples including well known operators are presented.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48583971","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}
A characterization of the generalized Lipschitz and Besov spaces in terms of decay of Fourier transforms is given. In particular, necessary and sufficient conditions of Titchmarsh type are obtained. The method is based on two-sided estimate for the rate of approximation of a $beta$-admissible family of multipliers operators in terms of decay properties of Fourier transform.
{"title":"Decay of Fourier transforms and generalized Besov spaces","authors":"T. Jordão","doi":"10.33205/cma.646557","DOIUrl":"https://doi.org/10.33205/cma.646557","url":null,"abstract":"A characterization of the generalized Lipschitz and Besov spaces in terms of decay of Fourier transforms is given. In particular, necessary and sufficient conditions of Titchmarsh type are obtained. The method is based on two-sided estimate for the rate of approximation of a $beta$-admissible family of multipliers operators in terms of decay properties of Fourier transform.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42621698","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 the paper is to investigate the approximation properties of bivariate generalization of Gauss-Weierstrass operators associated with the Riemann-Liouville operator. In particular, the approximation error will be estimated by these operators in the space of functions defined and continuous in the half-plane $(0, infty) times mathbb{R}$, and bounded by certain exponential functions.
本文的目的是研究与Riemann-Liouville算子相关的Gauss-Weierstrass算子的二元泛化的近似性质。特别地，逼近误差将由这些算子在半平面$(0, infty) times mathbb{R}$上连续定义的函数空间中估计，并以某些指数函数为界。
{"title":"On Some Bivariate Gauss-Weierstrass Operators","authors":"G. Krech, Ireneusz Krech","doi":"10.33205/CMA.518582","DOIUrl":"https://doi.org/10.33205/CMA.518582","url":null,"abstract":"The aim of the paper is to investigate the approximation properties of bivariate generalization of Gauss-Weierstrass operators associated with the Riemann-Liouville operator. In particular, the approximation error will be estimated by these operators in the space of functions defined and continuous in the half-plane $(0, infty) times mathbb{R}$, and bounded by certain exponential functions.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46087339","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 the classic result of Korovkin about the convergence of sequences of functions defined from sequences of linear operators is reformulated in terms of generalized convergence. This convergence extends some others given in the literature. The operator of the sequence fulfill a shape preserving property more general than the positivity. This property is related with certain extension of the notion of derivative. This extended derivative is precisely the object of the approximation process. The study is completed by analysing the conditions for the existence of an asymptotic formula, from which some interesting consequences are derived as a local version of the shape preserving property. Finally, as applications of the previous results, the author use the following notion of generalized convergence, an extension of Norlund-Cesaro summability given by V. Loku and N. L. Braha in 2017. A way to transfer a notion of generalized convergence to approximation theory by means of linear operators is showed .
{"title":"A General Korovkin Result Under Generalized Convergence","authors":"P. Garrancho","doi":"10.33205/CMA.530987","DOIUrl":"https://doi.org/10.33205/CMA.530987","url":null,"abstract":"In this paper the classic result of Korovkin about the convergence of sequences of functions defined from sequences of linear operators is reformulated in terms of generalized convergence. This convergence extends some others given in the literature. The operator of the sequence fulfill a shape preserving property more general than the positivity. This property is related with certain extension of the notion of derivative. This extended derivative is precisely the object of the approximation process. The study is completed by analysing the conditions for the existence of an asymptotic formula, from which some interesting consequences are derived as a local version of the shape preserving property. Finally, as applications of the previous results, the author use the following notion of generalized convergence, an extension of Norlund-Cesaro summability given by V. Loku and N. L. Braha in 2017. A way to transfer a notion of generalized convergence to approximation theory by means of linear operators is showed .","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47439329","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}
Choonkill Park, Sungsik Yun, Jung Rye Lee, D. Shin
In this paper, we introduce set-valued additive functional equations and prove the Hyers-Ulam stability of the set-valued additive functional equations by using the fixed point method.
本文引入了集值可加函数方程，并用不动点方法证明了集值加函数方程的Hyers-Ulam稳定性。
{"title":"Set-Valued Additive Functional Equations","authors":"Choonkill Park, Sungsik Yun, Jung Rye Lee, D. Shin","doi":"10.33205/CMA.528182","DOIUrl":"https://doi.org/10.33205/CMA.528182","url":null,"abstract":"In this paper, we introduce set-valued additive functional equations and prove the Hyers-Ulam stability of the set-valued additive functional equations by using the fixed point method.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46008169","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 study the existence and the norm of operators obtained as power series of linear positive operators with particularization to Bernstein operators. We also obtain a Voronovskaja-kind theorem.
{"title":"On Geometric Series of Positive Linear Operators","authors":"R. Păltănea","doi":"10.33205/CMA.506015","DOIUrl":"https://doi.org/10.33205/CMA.506015","url":null,"abstract":"We study the existence and the norm of operators obtained as power series of linear positive operators with particularization to Bernstein operators. We also obtain a Voronovskaja-kind theorem.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45720386","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}
Here we give a variety of general multivariate Iyengar type inequalities for not necessarily radial functions defined on the shell and ball. Our approach is based on the polar coordinates in $mathbb{R}^{N}$, $Ngeq 2$, and the related multivariate polar integration formula. Via this method we transfer well-known univariate Iyengar type inequalities and univariate author's related results into general multivariate Iyengar inequalities.
{"title":"General Multivariate Iyengar Type Inequalities","authors":"G. Anastassiou","doi":"10.33205/CMA.543560","DOIUrl":"https://doi.org/10.33205/CMA.543560","url":null,"abstract":"Here we give a variety of general multivariate Iyengar type inequalities for not necessarily radial functions defined on the shell and ball. Our approach is based on the polar coordinates in $mathbb{R}^{N}$, $Ngeq 2$, and the related multivariate polar integration formula. Via this method we transfer well-known univariate Iyengar type inequalities and univariate author's related results into general multivariate Iyengar inequalities.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41957537","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：