The Berezin transform $widetilde{A}$ and the Berezin radius of an operator $A$ on the reproducing kernel Hilbert space over some set $Q$ with normalized reproducing kernel $k_{eta}:=dfrac{K_{eta}}{leftVert K_{eta}rightVert}$ are defined, respectively, by $widetilde{A}(eta)=leftlangle {A}k_{eta},k_{eta}rightrangle$, $etain Q$ and $mathrm{ber} (A):=sup_{etain Q}leftvert widetilde{A}{(eta)}rightvert$. A simple comparison of these properties produces the inequalities $dfrac{1}{4}leftVert A^{ast}A+AA^{ast}rightVert leqmathrm{ber}^{2}left( Aright) leqdfrac{1}{2}leftVert A^{ast}A+AA^{ast}rightVert $. In this research, we investigate other inequalities that are related to them. In particular, for $Ainmathcal{L}left( mathcal{H}left(Qright) right) $ we prove that$mathrm{ber}^{2}left( Aright) leqdfrac{1}{2}leftVert A^{ast}A+AA^{ast}rightVert _{mathrm{ber}}-dfrac{1}{4}inf_{etain Q}left(left( widetilde{leftvert Arightvert }left( etaright)right)-left( widetilde{leftvert A^{ast}rightvert }left( etaright)right) right) ^{2}.$
{"title":"Improvements of some Berezin radius inequalities","authors":"M. Gürdal, M. Alomari","doi":"10.33205/cma.1110550","DOIUrl":"https://doi.org/10.33205/cma.1110550","url":null,"abstract":"The Berezin transform $widetilde{A}$ and the Berezin radius of an operator $A$ on the reproducing kernel Hilbert space over some set $Q$ with normalized reproducing kernel $k_{eta}:=dfrac{K_{eta}}{leftVert K_{eta}rightVert}$ are defined, respectively, by $widetilde{A}(eta)=leftlangle {A}k_{eta},k_{eta}rightrangle$, $etain Q$ and $mathrm{ber} (A):=sup_{etain Q}leftvert widetilde{A}{(eta)}rightvert$. A simple comparison of these properties produces the inequalities $dfrac{1}{4}leftVert A^{ast}A+AA^{ast}rightVert leqmathrm{ber}^{2}left( Aright) leqdfrac{1}{2}leftVert A^{ast}A+AA^{ast}rightVert $. In this research, we investigate other inequalities that are related to them. In particular, for $Ainmathcal{L}left( mathcal{H}left(Qright) right) $ we prove that$mathrm{ber}^{2}left( Aright) leqdfrac{1}{2}leftVert A^{ast}A+AA^{ast}rightVert _{mathrm{ber}}-dfrac{1}{4}inf_{etain Q}left(left( widetilde{leftvert Arightvert }left( etaright)right)-left( widetilde{leftvert A^{ast}rightvert }left( etaright)right) right) ^{2}.$","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48929216","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 rational meromorphic functions on $mathbb{C}backslashmathbb{R}$ are studied. We consider the some classes of one, as the generalized Nevanlinna $mathbf{N}_{kappa}$ and generalized Stieltjes $mathbf{N}_{kappa}^{k}$ classes. By Euclidean algorithm, we can find indices $kappa$ and $k$, i.e. determine which class the function belongs to $mathbf{N}_{kappa}^{k}$.
{"title":"Rational generalized Stieltjes functions","authors":"Professor DR.","doi":"10.33205/cma.1116322","DOIUrl":"https://doi.org/10.33205/cma.1116322","url":null,"abstract":"The rational meromorphic functions on $mathbb{C}backslashmathbb{R}$ are studied. We consider the some classes of one, as the generalized Nevanlinna $mathbf{N}_{kappa}$ and generalized Stieltjes $mathbf{N}_{kappa}^{k}$ classes. By Euclidean algorithm, we can find indices $kappa$ and $k$, i.e. determine which class the function belongs to $mathbf{N}_{kappa}^{k}$.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43440199","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 second and third powers of the Dirichlet kernel are used to construct discrete linear operators for the approximation of continuous periodic functions. An estimate of the rate of convergence is given. Approximation of non-periodic functions are also considered.
{"title":"Power of Dirichlet kernels and approximation by discrete linear operators {rm I}: direct results","authors":"J. Bustamante","doi":"10.33205/cma.1063594","DOIUrl":"https://doi.org/10.33205/cma.1063594","url":null,"abstract":"The second and third powers of the Dirichlet kernel are used to construct discrete linear operators for the approximation of continuous periodic functions. An estimate of the rate of convergence is given. Approximation of non-periodic functions are also considered.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49186082","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 concerns dual frames multipliers, i.e. operators in Hilbert spaces consisting of analysis, multiplication and synthesis processes, where the analysis and the synthesis are made by two dual frames, respectively. The goal of the paper is to give some results about the localization of the spectra of dual frames multipliers, i.e. to identify regions of the complex plane containing the spectra using some information about the frames and the symbols.
{"title":"Localization of the spectra of dual frames multipliers","authors":"R. Corso","doi":"10.33205/cma.1154703","DOIUrl":"https://doi.org/10.33205/cma.1154703","url":null,"abstract":"This paper concerns dual frames multipliers, i.e. operators in Hilbert spaces consisting of analysis, multiplication and synthesis processes, where the analysis and the synthesis are made by two dual frames, respectively. The goal of the paper is to give some results about the localization of the spectra of dual frames multipliers, i.e. to identify regions of the complex plane containing the spectra using some information about the frames and the symbols.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43263096","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 BSTRACT . A new definition of the incomplete beta function as a distribution-valued meromorphic function is given and the finite parts of it and of its partial derivatives at the singular values are calculated and compared with formulas in the literature.
{"title":"On the singular values of the incomplete Beta function","authors":"N. Ortner, P. Wagner","doi":"10.33205/cma.1086298","DOIUrl":"https://doi.org/10.33205/cma.1086298","url":null,"abstract":"A BSTRACT . A new definition of the incomplete beta function as a distribution-valued meromorphic function is given and the finite parts of it and of its partial derivatives at the singular values are calculated and compared with formulas in the literature.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46039896","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 $mathcal{M}$ be a semifinite von Neumann algebra on a Hilbert space $mathcal{H}$ equipped with a faithful normal semifinite trace $tau$, $S(mathcal{M},tau)$ be the ${}^*$-algebra of all $tau$-measurable operators. Let $S_0(mathcal{M},tau)$ be the ${}^*$-algebra of all $tau$-compact operators and $T(mathcal{M},tau)=S_0(mathcal{M},tau)+mathbb{C}I$ be the ${}^*$-algebra of all operators $X=A+lambda I$ with $Ain S_0(mathcal{M},tau)$ and $lambda in mathbb{C}$. It is proved that every operator of $T(mathcal{M},tau)$ that is left-invertible in $T(mathcal{M},tau)$ is in fact invertible in $T(mathcal{M},tau)$. It is a generalization of Sterling Berberian theorem (1982) on the subalgebra of thin operators in $mathcal{B} (mathcal{H})$. For the singular value function $mu(t; Q)$ of $Q=Q^2in S(mathcal{M},tau)$, the inclusion $mu(t; Q)in {0}bigcup [1, +infty)$ holds for all $t>0$. It gives the positive answer to the question posed by Daniyar Mushtari in 2010.
{"title":"The algebra of thin measurable operators is directly finite","authors":"A. Bikchentaev","doi":"10.33205/cma.1181495","DOIUrl":"https://doi.org/10.33205/cma.1181495","url":null,"abstract":"Let $mathcal{M}$ be a semifinite von Neumann algebra on a Hilbert space $mathcal{H}$ equipped with a faithful normal semifinite trace $tau$, $S(mathcal{M},tau)$ be the ${}^*$-algebra of all $tau$-measurable operators. Let $S_0(mathcal{M},tau)$ be the ${}^*$-algebra of all $tau$-compact operators and $T(mathcal{M},tau)=S_0(mathcal{M},tau)+mathbb{C}I$ be the ${}^*$-algebra of all operators $X=A+lambda I$\u0000 with $Ain S_0(mathcal{M},tau)$ and $lambda in mathbb{C}$. It is proved that every operator of $T(mathcal{M},tau)$ that is left-invertible in $T(mathcal{M},tau)$ is in fact invertible in $T(mathcal{M},tau)$.\u0000 It is a generalization of Sterling Berberian theorem (1982) on the subalgebra of thin operators in $mathcal{B} (mathcal{H})$.\u0000 For the singular value function $mu(t; Q)$ of $Q=Q^2in S(mathcal{M},tau)$, the inclusion $mu(t; Q)in {0}bigcup\u0000 [1, +infty)$ holds for all $t>0$. It gives the positive answer to the question posed by Daniyar Mushtari in 2010.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43053579","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}
{"title":"Parameters in Banach spaces and orthogonality","authors":"P. Papini, M. Baronti","doi":"10.33205/cma.1067323","DOIUrl":"https://doi.org/10.33205/cma.1067323","url":null,"abstract":"","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41464020","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}
Padua points, discovered in 2005 at the University of Padua, are the first set of points on the square [−1, 1]2 that are explicitly known, unisolvent for total degree polynomial interpolation and with Lebesgue constant increasing like log2(n) of the degree. One of the key features of the Padua Points is that they lie on a particular Lissajous curve. Other important properties of Padua points are 1. In two dimensions, Padua points are a WAM for interpolation and for extracting Approximate Fekete Points and Discrete Leja sequences. 2. In three dimensions, Padua points can be used for constructing tensor product WAMs on different compacts. Unfortunately their extension to higher dimensions is still the biggest open problem. The concept of mapped bases has been widely studied (cf. e.g. [35] and references therein), which turns out to be equivalent to map the interpolating nodes and then construct the approximant in the classical form without the need of resampling. The mapping technique is general, in the sense that works with any basis and can be applied to continuous, piecewise or discontinuous functions or even images. All the proposed methods show convergence to the interpolant provided that the function is resampled at the mapped nodes. In applications, this is often physically unfeasible. An effective method for interpolating via mapped bases in the multivariate setting, referred as Fake Nodes Approach (FNA), has been presented in [38]. In this paper, some interesting connection of the FNA with Padua points and “families of relatives nodes”, that can be used as “fake nodes” for multivariate approximation, are presented and we conclude with some open problems.
{"title":"Padua points and fake nodes for polynomial approximation: old, new and open problems","authors":"S. De Marchi","doi":"10.33205/cma.1070020","DOIUrl":"https://doi.org/10.33205/cma.1070020","url":null,"abstract":"Padua points, discovered in 2005 at the University of Padua, are the first set of points on the square [−1, 1]2 that are explicitly known, unisolvent for total degree polynomial interpolation and with Lebesgue constant increasing like log2(n) of the degree. One of the key features of the Padua Points is that they lie on a particular Lissajous curve. Other important properties of Padua points are 1. In two dimensions, Padua points are a WAM for interpolation and for extracting Approximate Fekete Points and Discrete Leja sequences. 2. In three dimensions, Padua points can be used for constructing tensor product WAMs on different compacts. Unfortunately their extension to higher dimensions is still the biggest open problem. The concept of mapped bases has been widely studied (cf. e.g. [35] and references therein), which turns out to be equivalent to map the interpolating nodes and then construct the approximant in the classical form without the need of resampling. The mapping technique is general, in the sense that works with any basis and can be applied to continuous, piecewise or discontinuous functions or even images. All the proposed methods show convergence to the interpolant provided that the function is resampled at the mapped nodes. In applications, this is often physically unfeasible. An effective method for interpolating via mapped bases in the multivariate setting, referred as Fake Nodes Approach (FNA), has been presented in [38]. In this paper, some interesting connection of the FNA with Padua points and “families of relatives nodes”, that can be used as “fake nodes” for multivariate approximation, are presented and we conclude with some open problems.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44125881","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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