This paper revisits the Gneiting class of positive definite kernels originally proposed as a class of covariance functions for space-time processes. Under the framework of quasi-metric spaces and isometric embeddings, the paper proposes a general and unifying framework that encompasses results provided by earlier literature. Our results allow to study the positive definiteness of the Gneiting class over products of either Euclidean spaces or high dimensional spheres and quasi-metric spaces. In turn, Gneiting's theorem is proved here by a direct construction, eluding Fourier inversion (the so-called Gneiting's lemma) and convergence arguments that are required by Gneiting to preserve an integrability assumption.
{"title":"Gneiting Class, Semi-Metric Spaces and Isometric Embeddings","authors":"V. Menegatto, C. Oliveira, E. Porcu","doi":"10.33205/cma.712049","DOIUrl":"https://doi.org/10.33205/cma.712049","url":null,"abstract":"This paper revisits the Gneiting class of positive definite kernels originally proposed as a class of covariance functions for space-time processes. Under the framework of quasi-metric spaces and isometric embeddings, the paper proposes a general and unifying framework that encompasses results provided by earlier literature. Our results allow to study the positive definiteness of the Gneiting class over products of either Euclidean spaces or high dimensional spheres and quasi-metric spaces. In turn, Gneiting's theorem is proved here by a direct construction, eluding Fourier inversion (the so-called Gneiting's lemma) and convergence arguments that are required by Gneiting to preserve an integrability assumption.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44623720","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}
Our results concern growth estimates for vector-valued functions of $mathbb{L}$-index in joint variables which are analytic in the unit ball. There are deduced analogs of known growth estimates obtained early for functions analytic in the unit ball. Our estimates contain logarithm of $sup$-norm instead of logarithm modulus of the function. They describe the behavior of logarithm of norm of analytic vector-valued function on a skeleton in a bidisc by behavior of the function $mathbf{L}.$ These estimates are sharp in a general case. The presented results are based on bidisc exhaustion of a unit ball.
{"title":"Growth Estimates for Analytic Vector-Valued Functions in the Unit Ball Having Bounded $mathbf{L}$-index in Joint Variables","authors":"V. Baksa, Andriy Ivanovych Bandura, O. Skaskiv","doi":"10.33205/CMA.650977","DOIUrl":"https://doi.org/10.33205/CMA.650977","url":null,"abstract":"Our results concern growth estimates for vector-valued functions of $mathbb{L}$-index in joint variables which are analytic in the unit ball. There are deduced analogs of known growth estimates obtained early for functions analytic in the unit ball. Our estimates contain logarithm of $sup$-norm instead of logarithm modulus of the function. They describe the behavior of logarithm of norm of analytic vector-valued function on a skeleton in a bidisc by behavior of the function $mathbf{L}.$ These estimates are sharp in a general case. The presented results are based on bidisc exhaustion of a unit ball.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45783216","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 study the following family of hypergeometric polynomials: $y_n(x) = frac{ (-1)^rho }{ n! } x^n {}_2 F_0(-n,rho;-;-frac{1}{x})$, depending on a parameter $rhoinmathbb{N}$. Differential equations of orders $rho+1$ and $2$ for these polynomials are given. A recurrence relation for $y_n$ is derived as well. Polynomials $y_n$ are Sobolev orthogonal polynomials on the unit circle with an explicit matrix measure.
{"title":"On a Family of Hypergeometric Sobolev Orthogonal Polynomials on the Unit Circle","authors":"S. Zagorodnyuk","doi":"10.33205/cma.690236","DOIUrl":"https://doi.org/10.33205/cma.690236","url":null,"abstract":"In this paper we study the following family of hypergeometric polynomials: $y_n(x) = frac{ (-1)^rho }{ n! } x^n {}_2 F_0(-n,rho;-;-frac{1}{x})$, depending on a parameter $rhoinmathbb{N}$. Differential equations of orders $rho+1$ and $2$ for these polynomials are given. A recurrence relation for $y_n$ is derived as well. Polynomials $y_n$ are Sobolev orthogonal polynomials on the unit circle with an explicit matrix measure.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43487318","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 summation/integration method for fast summing trigonometric series is presented. The basic idea in this method is to transform the series to an integral with respect to some weight function on $RR_+$ and then to approximate such an integral by the appropriate quadrature formulas of Gaussian type. The construction of these quadrature rules, as well as the corresponding orthogonal polynomials on $RR_+$, are also considered. Finally, in order to illustrate the efficiency of the presented summation/integration method two numerical examples are included.
{"title":"Quadrature Formulas of Gaussian Type for Fast Summation of Trigonometric Series","authors":"G. Milovanović","doi":"10.33205/cma.613948","DOIUrl":"https://doi.org/10.33205/cma.613948","url":null,"abstract":"A summation/integration method for fast summing trigonometric series is presented. The basic idea in this method is to transform the series to an integral with respect to some weight function on $RR_+$ and then to approximate such an integral by the appropriate quadrature formulas of Gaussian type. The construction of these quadrature rules, as well as the corresponding orthogonal polynomials on $RR_+$, are also considered. Finally, in order to illustrate the efficiency of the presented summation/integration method two numerical examples are included.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43907942","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}
Beals, Gaveau, and Greiner established a formula for the fundamental solution to the Laplace equation with drift term in Grushin-type planes. The first author and Childers expanded these results by invoking a p-Laplace type generalization that encompasses these formulas while the authors explored a different natural generalization of the p-Laplace equation with drift term that also encompasses these formulas. In both, the drift term lies in the complex domain. We extend these results by considering a drift term in the quaternion realm and show our solutions are stable under limits as p tends to infinity.
{"title":"Generalizations of the drift Laplace equation over the quaternions in a class of Grushin-type spaces","authors":"Thomas Bieske, Keller Blackwell","doi":"10.33205/cma.1324774","DOIUrl":"https://doi.org/10.33205/cma.1324774","url":null,"abstract":"Beals, Gaveau, and Greiner established a formula for the fundamental solution to the Laplace equation with drift term in Grushin-type planes. The first author and Childers expanded these results by invoking a p-Laplace type generalization that encompasses these formulas while the authors explored a different natural generalization of the p-Laplace equation with drift term that also encompasses these formulas. In both, the drift term lies in the complex domain. We extend these results by considering a drift term in the quaternion realm and show our solutions are stable under limits as p tends to infinity.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46597067","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 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":null,"pages":null},"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}
The present note is devoted to a generalization of the notion of shift invariant operators that we call it $lambda $-invariant operators $(lambda ge 0)$. Some properties of this new class are presented. By using probabilistic methods, three examples are delivered.
本笔记致力于移位不变量算子概念的推广，我们称之为$lambda $ -不变量算子$(lambda ge 0)$。给出了该类的一些性质。利用概率方法，给出了三个实例。
{"title":"Shift $lambda $-Invariant Operators","authors":"O. Agratini","doi":"10.33205/CMA.544094","DOIUrl":"https://doi.org/10.33205/CMA.544094","url":null,"abstract":"The present note is devoted to a generalization of the notion of shift invariant operators that we call it $lambda $-invariant operators $(lambda ge 0)$. Some properties of this new class are presented. By using probabilistic methods, three examples are delivered.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47693814","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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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：