Recently, the construction of 2D empirical wavelets based on partitioning the Fourier domain with the watershed transform has been proposed. If such approach can build partitions of completely arbitrary shapes, for some applications, it is desirable to keep a certain level of regularity in the geometry of the obtained partitions. In this paper, we propose to build such partition using Voronoi diagrams. This solution allows us to keep a high level of adaptability while guaranteeing a minimum level of geometric regularity in the detected partition.
{"title":"Empirical Voronoi wavelets","authors":"J. Gilles","doi":"10.33205/cma.1181174","DOIUrl":"https://doi.org/10.33205/cma.1181174","url":null,"abstract":"Recently, the construction of 2D empirical wavelets based on partitioning the Fourier domain with the watershed transform has been proposed. If such approach can build partitions of completely arbitrary shapes, for some applications, it is desirable to keep a certain level of regularity in the geometry of the obtained partitions. In this paper, we propose to build such partition using Voronoi diagrams. This solution allows us to keep a high level of adaptability while guaranteeing a minimum level of geometric regularity in the detected partition.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41616100","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 construct a sampling operator with the property that the smoother a function is, the faster its approximation is. We establish a direct estimate and a weak converse estimate of its rate of approximation in the uniform norm by means of a modulus of smoothness and a $K$-functional. The case of weighted approximation is also considered. The weights are positive and power-type with non-positive exponents at infinity. This sampling operator preserves every algebraic polynomial.
{"title":"A fast converging sampling operator","authors":"B. Draganov","doi":"10.33205/cma.1172005","DOIUrl":"https://doi.org/10.33205/cma.1172005","url":null,"abstract":"We construct a sampling operator with the property that the smoother a function is, the faster its approximation is. We establish a direct estimate and a weak converse estimate of its rate of approximation in the uniform norm by means of a modulus of smoothness and a $K$-functional. The case of weighted approximation is also considered. The weights are positive and power-type with non-positive exponents at infinity. This sampling operator preserves every algebraic polynomial.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43170311","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}
As one new result, for a symmetric Toeplitz $ operatorname{sinc} $ $n times n$-matrix $A(t)$ depending on a parameter $t$, lower estimates (tending to infinity as t vanishes) on the pertinent condition number are derived. A further important finding is that prior to improving the obtained lower estimates it seems to be more important to determine the lower bound on the parameter $t$ such that the smallest eigenvalue $mu_n(t)$ of $A(t)$ can be reliably computed since this is a precondition for determining a reliable value for the condition number of the Toeplitz $ operatorname{sinc} $ matrix. The style of the paper is expository in order to address a large readership.
{"title":"Lower estimates on the condition number of a Toeplitz sinc matrix and related questions","authors":"L. Kohaupt, Yan Wu","doi":"10.33205/cma.1142905","DOIUrl":"https://doi.org/10.33205/cma.1142905","url":null,"abstract":"As one new result, for a symmetric Toeplitz $ operatorname{sinc} $ $n times n$-matrix $A(t)$ depending on a parameter $t$, lower estimates (tending to infinity as t vanishes) on the pertinent condition number are derived. A further important finding is that prior to improving the obtained lower estimates it seems to be more important to determine the lower bound on the parameter $t$ such that the smallest eigenvalue $mu_n(t)$ of $A(t)$ can be reliably computed since this is a precondition for determining a reliable value for the condition number of the Toeplitz $ operatorname{sinc} $ matrix. The style of the paper is expository in order to address a large readership.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48398427","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 construct a solution of the Poisson equation in exterior domains $Omega subset mathbb R^n,;n ge 2,$ in homogeneous Lebesgue spaces $L^{2,q}(Omega),;1 < q
利用势理论和积分方程的方法构造齐次Lebesgue空间$L^{2,q}(Omega),;1 < q
{"title":"On the Poisson equation in exterior domains","authors":"W. Varnhorn","doi":"10.33205/cma.1143800","DOIUrl":"https://doi.org/10.33205/cma.1143800","url":null,"abstract":"We construct a solution of the Poisson equation in exterior domains $Omega subset mathbb R^n,;n ge 2,$ in homogeneous Lebesgue spaces $L^{2,q}(Omega),;1 < q <infty,$ with methods of potential theory and integral equations. We investigate the corresponding null spaces and prove that its dimensions is equal to $n+1$ independent of $q$.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46709038","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 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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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}
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