We present uniform and $L_p$ mixed Caputo-Bochner abstract generalized fractional Landau inequalities over $mathbb{R}$ of fractional orders $ 2 < alpha leq 3 $. These estimate the size of first and second derivatives of a composition with a Banach space valued function over $mathbb{R}$. We give applications when $α = 2.5$.
{"title":"Abstract Generalized Fractional Landau inequalities over R","authors":"G. Anastassiou","doi":"10.33205/CMA.764161","DOIUrl":"https://doi.org/10.33205/CMA.764161","url":null,"abstract":"We present uniform and $L_p$ mixed Caputo-Bochner abstract generalized fractional Landau inequalities over $mathbb{R}$ of fractional orders $ 2 < alpha leq 3 $. These estimate the size of first and second derivatives of a composition with a Banach space valued function over $mathbb{R}$. We give applications when $α = 2.5$.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48991874","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}
It is known that the restricted Riesz transform of a Lebesgue integrable function is not Lebesgue integrable. In this paper we prove that the restricted Riesz transform of a Lebesgue integrable function is A-integrable and the analogue of Riesz's equality holds. ABSTRACT.It is known that the restricted Riesz transform of a Lebesgue integrable function is not Lebesgue inte-grable. In this paper, we prove that the restricted Riesz transform of a Lebesgue integrable function isA-integrableand the analogue of Riesz’s equality holds
{"title":"The A-integral and restricted Riesz transform","authors":"R. Aliev, Khanim I. Nebiyeva","doi":"10.33205/cma.728156","DOIUrl":"https://doi.org/10.33205/cma.728156","url":null,"abstract":"It is known that the restricted Riesz transform of a Lebesgue integrable function is not Lebesgue integrable. In this paper we prove that the restricted Riesz transform of a Lebesgue integrable function is A-integrable and the analogue of Riesz's equality holds. ABSTRACT.It is known that the restricted Riesz transform of a Lebesgue integrable function is not Lebesgue inte-grable. In this paper, we prove that the restricted Riesz transform of a Lebesgue integrable function isA-integrableand the analogue of Riesz’s equality holds","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46431542","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}
Roughly speaking an equation is called Ulam stable if near each approximate solution of the equation there exists an exact solution. In this paper we prove that Cauchy-Schwarz equation, Ortogonality equation and Gram equation are Ulam stable. This paper is concerned with the Ulam stability of some classical equations arising in thecontext of inner-product spaces. For the general notion of Ulam stability see, e.q., [1]. Roughlyspeaking an equation is called Ulam stable if near every approximate solution there exists anexact solution; the precise meaning in each case presented in this paper is described in threetheorems. Related results can be found in [2, 3, 4]. See also [5] for some inequalities in innerproduct spaces.
{"title":"Ulam stability in real inner-product spaces","authors":"Bianca Moșneguțu, A. Mǎdutǎ","doi":"10.33205/cma.758854","DOIUrl":"https://doi.org/10.33205/cma.758854","url":null,"abstract":"Roughly speaking an equation is called Ulam stable if near each approximate solution of the equation there exists an exact solution. In this paper we prove that Cauchy-Schwarz equation, Ortogonality equation and Gram equation are Ulam stable. This paper is concerned with the Ulam stability of some classical equations arising in thecontext of inner-product spaces. For the general notion of Ulam stability see, e.q., [1]. Roughlyspeaking an equation is called Ulam stable if near every approximate solution there exists anexact solution; the precise meaning in each case presented in this paper is described in threetheorems. Related results can be found in [2, 3, 4]. See also [5] for some inequalities in innerproduct spaces.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48370366","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}
Let $sigma_n$ denotes the classical Fej'er operator for trigonometric expansions. For a fixed even integer $r$, we characterize the rate of convergence of the iterative operators $(I-sigma_n)^r(f)$ in terms of the modulus of continuity of order $r$ (with specific constants) in all $mathbb{L}^p$ spaces $1leq p leq infty$. In particular, the constants depend not on $p$. Moreover, we present a quantitative version of the Voronovskaya-type theorems for the operators $(I-sigma_n)^r(f)$.
{"title":"Strong converse inequalities and quantitative Voronovskaya-type theorems for trigonometric Fej'er sums","authors":"J. Bustamante, Lázaro Flores De Jesús","doi":"10.33205/cma.653843","DOIUrl":"https://doi.org/10.33205/cma.653843","url":null,"abstract":"Let $sigma_n$ denotes the classical Fej'er operator for trigonometric expansions. For a fixed even integer $r$, we characterize the rate of convergence of the iterative operators $(I-sigma_n)^r(f)$ in terms of the modulus of continuity of order $r$ (with specific constants) in all $mathbb{L}^p$ spaces $1leq p leq infty$. In particular, the constants depend not on $p$. Moreover, we present a quantitative version of the Voronovskaya-type theorems for the operators $(I-sigma_n)^r(f)$.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48490943","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}
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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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}
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":" ","pages":""},"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}
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