{"title":"Isomorphism problem in a special class of Banach function algebras and its application","authors":"Kiyoshi Shirayanagi","doi":"10.33205/cma.952056","DOIUrl":"https://doi.org/10.33205/cma.952056","url":null,"abstract":"","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41678321","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 extend the classical van der Corput inequality to the real line. As a consequence, we obtain a simple proof of the Wiener-Wintner theorem for the $mathbb{R}$-action which assert that for any family of maps $(T_t)_{t in mathbb{R}}$ acting on the Lebesgue measure space $(Omega,{cal {A}},mu)$ where $mu$ is a probability measure and for any $tin mathbb{R}$, $T_t$ is measure-preserving transformation on measure space $(Omega,{cal {A}},mu)$ with $T_t circ T_s =T_{t+s}$, for any $t,sin mathbb{R}$. Then, for any $f in L^1(mu)$, there is a a single null set off which $displaystyle lim_{T rightarrow +infty} frac1{T}int_{0}^{T} f(T_tomega) e^{2 i pi theta t} dt$ exists for all $theta in mathbb{R}$. We further present the joining proof of the amenable group version of Wiener-Wintner theorem due to Weiss and Ornstein.
{"title":"van der Corput inequality for real line and Wiener-Wintner theorem for amenable groups","authors":"E. Abdalaoui","doi":"10.33205/cma.1029202","DOIUrl":"https://doi.org/10.33205/cma.1029202","url":null,"abstract":"We extend the classical van der Corput inequality to the real line. As a consequence, we obtain a simple proof of the Wiener-Wintner theorem for the $mathbb{R}$-action which assert that for any family of maps $(T_t)_{t in mathbb{R}}$ acting on the Lebesgue measure space $(Omega,{cal {A}},mu)$ where $mu$ is a probability measure and for any $tin mathbb{R}$, $T_t$ is measure-preserving transformation on measure space $(Omega,{cal {A}},mu)$ with $T_t circ T_s =T_{t+s}$, for any $t,sin mathbb{R}$. Then, for any $f in L^1(mu)$, there is a a single null set off which $displaystyle lim_{T rightarrow +infty} frac1{T}int_{0}^{T} f(T_tomega) e^{2 i pi theta t} dt$ exists for all $theta in mathbb{R}$. We further present the joining proof of the amenable group version of Wiener-Wintner theorem due to Weiss and Ornstein.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48384265","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 introduce a method to construct general multivariate positive definite kernels on a nonempty set X that employs a prescribed bounded completely monotone function and special multivariate functions on X. The method is consistent with a generalized version of Aitken’s integral formula for Gaussians. In the case where X is a cartesian product, the method produces nonseparable positive definite kernels that may be useful in multivariate interpolation. In addition, it can be interpreted as an abstract multivariate generalization of the well-established Gneiting’s model for constructing space-time covariances commonly cited in the literature. Many parametric models discussed in statistics can be interpreted as particular cases of the method.
{"title":"Matrix valued positive definite kernels related to the generalized Aitken's integral for Gaussians","authors":"V. Menegatto, C. P. Oliveira","doi":"10.33205/cma.964096","DOIUrl":"https://doi.org/10.33205/cma.964096","url":null,"abstract":"We introduce a method to construct general multivariate positive definite kernels on a nonempty set X that employs a prescribed bounded completely monotone function and special multivariate functions on X. The method is consistent with a generalized version of Aitken’s integral formula for Gaussians. In the case where X is a cartesian product, the method produces nonseparable positive definite kernels that may be useful in multivariate interpolation. In addition, it can be interpreted as an abstract multivariate generalization of the well-established Gneiting’s model for constructing space-time covariances commonly cited in the literature. Many parametric models discussed in statistics can be interpreted as particular cases of the method.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41673806","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 present the duality theory for general weighted space of vector functions. We mention that a characterization of the dual of a weighted space of vector functions in the particular case $V subset C^{+} (X)$ is mentioned by J. B. Prolla in [6]. Also, we extend de Branges lemma in this new setting for convex cones of a weighted spaces of vector functions (Theorem 4.2). Using this theorem, we find various approximations results for weighted spaces of vector functions: Theorems 4.2-4.6 as well as Corollary 4.3. We mention also that a brief version of this paper, in the particular case $V subset C^{+} (X)$, is presented in [3], Chapter 2, subparagraph 2.5.
本文给出了广义向量函数加权空间的对偶理论。我们提到了J. B. Prolla在[6]中提到的在特殊情况下向量函数加权空间的对偶的一个表征$V 子集C^{+} (X)$。同时，我们在向量函数加权空间的凸锥的这种新设置中推广了de Branges引理(定理4.2)。利用这个定理，我们得到了向量函数加权空间的各种近似结果:定理4.2-4.6和推论4.3。我们还提到，本文的一个简短版本，在特殊情况下$V 子集C^{+} (X)$，在[3]，第2章，分段2.5中给出。
{"title":"APROXIMATION IN WEIGHTED SPACES OF VECTOR FUNCTIONS","authors":"G. Păltineanu, I. Bucur","doi":"10.33205/CMA.825986","DOIUrl":"https://doi.org/10.33205/CMA.825986","url":null,"abstract":"In this paper, we present the duality theory for general weighted space of vector functions. We mention that a characterization of the dual of a weighted space of vector functions in the particular case $V subset C^{+} (X)$ is mentioned by J. B. Prolla in [6]. Also, we extend de Branges lemma in this new setting for convex cones of a weighted spaces of vector functions (Theorem 4.2). Using this theorem, we find various approximations results for weighted spaces of vector functions: Theorems 4.2-4.6 as well as Corollary 4.3. We mention also that a brief version of this paper, in the particular case $V subset C^{+} (X)$, is presented in [3], Chapter 2, subparagraph 2.5.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69532123","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}
L. Angeloni, N. Çetin, D. Costarelli, A. R. Sambucini, G. Vinti
In this paper, we establish a quantitative estimate for multivariate sampling Kantorovich operators by means of the modulus of continuity in the general setting of Orlicz spaces. As a consequence, the qualitative order of convergence can be obtained, in case of functions belonging to suitable Lipschitz classes. In the particular instance of L^p-spaces, using a direct approach, we obtain a sharper estimate than that one that can be deduced from the general case.
{"title":"Multivariate sampling Kantorovich operators: quantitative estimates in Orlicz spaces","authors":"L. Angeloni, N. Çetin, D. Costarelli, A. R. Sambucini, G. Vinti","doi":"10.33205/CMA.876890","DOIUrl":"https://doi.org/10.33205/CMA.876890","url":null,"abstract":"In this paper, we establish a quantitative estimate for multivariate sampling Kantorovich operators by means of the modulus of continuity in the general setting of Orlicz spaces. As a consequence, the qualitative order of convergence can be obtained, in case of functions belonging to suitable Lipschitz classes. In the particular instance of L^p-spaces, using a direct approach, we obtain a sharper estimate than that one that can be deduced from the general case.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48360133","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 some recent applications of the so-called Generalized Bernstein polynomials are collected. This polynomial sequence is constructed by means of the samples of a continuous function f on equispaced points of [0; 1] and depends on an additional parameter which yields the remarkable property of improving the rate of convergence to the function f, according with the smoothness of f. This means that the sequence does not suffer of the saturation phenomena occurring by using the classical Bernstein polynomials or arising in piecewise polynomial approximation. The applications considered here deal with the numerical integration and the simultaneous approximation. Quadrature rules on equidistant nodes of [0; 1] are studied for the numerical computation of ordinary integrals in one or two dimensions, and usefully employed in Nystrom methods for solving Fredholm integral equations. Moreover, the simultaneous approximation of the Hilbert transform and its derivative (the Hadamard transform) is illustrated. For all the applications, some numerical details are given in addition to the error estimates, and the proposed approximation methods have been implemented providing numerical tests which confirm the theoretical estimates. Some open problems are also introduced.
{"title":"Some numerical applications of generalized Bernstein operators","authors":"D. Occorsio, M. Russo, W. Themistoclakis","doi":"10.33205/CMA.868272","DOIUrl":"https://doi.org/10.33205/CMA.868272","url":null,"abstract":"In this paper some recent applications of the so-called Generalized Bernstein polynomials are collected. This polynomial sequence is constructed by means of the samples of a continuous function f on equispaced points of [0; 1] and depends on an additional parameter which yields the remarkable property of improving the rate of convergence to the function f, according with the smoothness of f. This means that the sequence does not suffer of the saturation phenomena occurring by using the classical Bernstein polynomials or arising in piecewise polynomial approximation. The applications considered here deal with the numerical integration and the simultaneous approximation. Quadrature rules on equidistant nodes of [0; 1] are studied for the numerical computation of ordinary integrals in one or two dimensions, and usefully employed in Nystrom methods for solving Fredholm integral equations. Moreover, the simultaneous approximation of the Hilbert transform and its derivative (the Hadamard transform) is illustrated. For all the applications, some numerical details are given in addition to the error estimates, and the proposed approximation methods have been implemented providing numerical tests which confirm the theoretical estimates. Some open problems are also introduced.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41487043","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}
Pub Date : 2021-02-01DOI: 10.1142/9789811221644_0001
{"title":"Metric Spaces and Limits for Sequences","authors":"","doi":"10.1142/9789811221644_0001","DOIUrl":"https://doi.org/10.1142/9789811221644_0001","url":null,"abstract":"","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86547024","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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