{"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":"1 1","pages":""},"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":" ","pages":""},"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}
{"title":"Norm attaining multilinear forms on the spaces $c_0$ or $l_1$","authors":"Sung Guen Kim","doi":"10.33205/cma.981877","DOIUrl":"https://doi.org/10.33205/cma.981877","url":null,"abstract":"","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48012114","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":"Oscillation of noncanonical second-order advanced differential equations via canonical transform","authors":"M. Bohner, K. Vidhyaa, E. Thandapani","doi":"10.33205/cma.1055356","DOIUrl":"https://doi.org/10.33205/cma.1055356","url":null,"abstract":"","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45317439","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}
Abstract. Hilberdink showed that there exists a constant c0 > 2, such that there exists a continuous prim system satisfying N(x) = c(x − 1) + 1 if and only if c ≤ c0. Here we determine c0 numerically to be 1.25479 ·10 ±2 ·10 . To do so we compute a representation for a twisted exponential function as a sum over the roots of the Riemann zeta function. We then give explicit bounds for the error obtained when restricting the occurring sum to a finite number of zeros.
{"title":"Continuous prime systems satisfying N(x)=c(x-1)+1","authors":"J. Schlage-Puchta","doi":"10.33205/cma.817761","DOIUrl":"https://doi.org/10.33205/cma.817761","url":null,"abstract":"Abstract. Hilberdink showed that there exists a constant c0 > 2, such that there exists a continuous prim system satisfying N(x) = c(x − 1) + 1 if and only if c ≤ c0. Here we determine c0 numerically to be 1.25479 ·10 ±2 ·10 . To do so we compute a representation for a twisted exponential function as a sum over the roots of the Riemann zeta function. We then give explicit bounds for the error obtained when restricting the occurring sum to a finite number of zeros.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43259612","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 consider the set of monic real uni-variate polynomials of a given degree $d$ with non-vanishing coefficients, with given signs of the coefficients and with given quantities $pos$ of their positive and $neg$ of their negative roots (all roots are distinct). For $dgeq 6$ and for signs of the coefficients $(+,-,+,+,ldots ,+,+,-,+)$, we prove that the set of such polynomials having two positive, $d-4$ negative and two complex conjugate roots, is not connected. For $pos+negleq 3$ and for any $d$, we give the exhaustive answer to the question for which signs of the coefficients there exist polynomials with such values of $pos$ and $neg$.
{"title":"The disconnectedness of certain sets defined after uni-variate polynomials","authors":"V. Kostov","doi":"10.33205/cma.1111247","DOIUrl":"https://doi.org/10.33205/cma.1111247","url":null,"abstract":"We consider the set of monic real uni-variate polynomials of a given degree $d$ with non-vanishing coefficients, with given signs of the coefficients and with given quantities $pos$ of their positive and $neg$ of their negative roots (all roots are distinct). For $dgeq 6$ and for signs of the coefficients $(+,-,+,+,ldots ,+,+,-,+)$, we prove that the set of such polynomials having two positive, $d-4$ negative and two complex conjugate roots, is not connected. For $pos+negleq 3$ and for any $d$, we give the exhaustive answer to the question for which signs of the coefficients there exist polynomials with such values of $pos$ and $neg$.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45784855","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":"Differential1 ${e}$-structures for equivalences of $2$-nondegenerate Levi rank $1$ hypersurfaces $M_5 ⊂ C$","authors":"W. Foo, J. Merker","doi":"10.33205/cma.943426","DOIUrl":"https://doi.org/10.33205/cma.943426","url":null,"abstract":"","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49295584","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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