Rational functions have deep system-theoretic significance. They represent the natural way of modeling linear dynamical systems in the frequency (Laplace) domain. Using rational functions, the goal of this paper to compute models that match (interpolate) given data sets of measurements. In this paper, the authors show that using special rational orthonormal systems, the Malmquist-Takenaka systems, it is possible to write the rational interpolant $r_{(n, m)}$, for $n=N-1, m=N$ using only $N$ sampling nodes (instead of $2N$ nodes) if the interpolating nodes are in the complex unit circle or on the upper half-plane. Moreover, the authors prove convergence results related to the rational interpolant. They give an efficient algorithm for the determination of the rational interpolant.
有理函数具有深刻的系统理论意义。它们代表了在频率(拉普拉斯)域中对线性动力系统进行建模的自然方法。使用有理函数，本文的目标是计算匹配(插值)给定测量数据集的模型。本文证明了在特殊的有理正交系统Malmquist-Takenaka系统中，对于$n= n -1, m= n $的有理插值$r_{(n, m)}$，只要使用$n $采样节点(而不是$2N$节点)就可以写出$n= 1, m= n $的有理插值$r_{(n, m)}$，如果插值节点位于复单位圆或上半平面上。此外，作者还证明了有关有理插值的收敛性结果。给出了一种确定有理插值的有效算法。
{"title":"Construction of rational interpolations using Mamquist-Takenaka systems","authors":"F. Weisz","doi":"10.33205/cma.1251068","DOIUrl":"https://doi.org/10.33205/cma.1251068","url":null,"abstract":"Rational functions have deep system-theoretic significance. They represent the natural way of modeling linear dynamical systems in the frequency (Laplace) domain. Using rational functions, the goal of this paper to compute models that match (interpolate) given data sets of measurements. In this paper, the authors show that using special rational orthonormal systems, the Malmquist-Takenaka systems, it is possible to write the rational interpolant $r_{(n, m)}$, for $n=N-1, m=N$ using only $N$ sampling nodes (instead of $2N$ nodes) if the interpolating nodes are in the complex unit circle or on the upper half-plane. Moreover, the authors prove convergence results related to the rational interpolant. They give an efficient algorithm for the determination of the rational interpolant.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48133956","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 derive some branched continued fraction representations for the ratios of the Horn's confluent function $mathrm{H}_6.$ The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. We establish the estimates of the rate of convergence for the branched continued fraction expansions in some region $Omega$ (here, region is a domain (open connected set) together with all, part or none of its boundary). It is also proved that the corresponding branched continued fractions uniformly converge to holomorphic functions on every compact subset of some domain $Theta,$ and that these functions are analytic continuations of the ratios of double confluent hypergeometric series in $Theta.$ At the end, several numerical experiments are represented to indicate the power and efficiency of branched continued fractions as an approximation tool compared to double confluent hypergeometric series.
{"title":"Branched continued fraction representations of ratios of Horn's confluent function $mathrm{H}_6$","authors":"T. Antonova, R. Dmytryshyn, S. Sharyn","doi":"10.33205/cma.1243021","DOIUrl":"https://doi.org/10.33205/cma.1243021","url":null,"abstract":"In this paper, we derive some branched continued fraction representations for the ratios of the Horn's confluent function $mathrm{H}_6.$ The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. We establish the estimates of the rate of convergence for the branched continued fraction expansions in some region $Omega$ (here, region is a domain (open connected set) together with all, part or none of its boundary). It is also proved that the corresponding branched continued fractions uniformly converge to holomorphic functions on every compact subset of some domain $Theta,$ and that these functions are analytic continuations of the ratios of double confluent hypergeometric series in $Theta.$ At the end, several numerical experiments are represented to indicate the power and efficiency of branched continued fractions as an approximation tool compared to double confluent hypergeometric series.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47890760","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 article, we study the space $mathcal B_mu(B_X,Y)$ of $Y$-valued Bloch-type functions on the unit ball $B_X$ of an infinite dimensional Hilbert space $X$ with $mu$ is a normal weight on $B_X$ and $Y$ is a Banach space. We also investigate the characterizations of the space $mathcal{WB}_mu(B_X)$ of $Y$-valued, locally bounded, weakly holomorphic functions associated with the Bloch-type space $mathcal B_mu(B_X)$ of scalar-valued functions in the sense that $fin mathcal{WB}_mu(B_X)$ if $wcirc f in mathcal B_mu(B_X)$ for every $w in mathcal W,$ a separating subspace of the dual $Y'$ of $Y.$
本文研究了无穷维Hilbert空间$X$的单位球$B_X$上的$Y$值Bloch型函数的空间$mathcal B_。我们还研究了空间$mathcal的特征{WB}_$Y$的μ（B_X）$—与标量值函数的Bloch型空间$mathcal B_mu（B_X{WB}_mu（B_X）$如果$wcirc finmathcal B_mu（B_X）$对于每$winmath cal w，$Y的对偶$Y'$的分离子空间$
{"title":"Banach-valued Bloch-type functions on the unit ball of a Hilbert space and weak spaces of Bloch-type","authors":"T. Quang","doi":"10.33205/cma.1243686","DOIUrl":"https://doi.org/10.33205/cma.1243686","url":null,"abstract":"In this article, we study the space $mathcal B_mu(B_X,Y)$ of $Y$-valued Bloch-type functions on the unit ball $B_X$ of an infinite dimensional Hilbert space $X$ with $mu$ is a normal weight on $B_X$ and $Y$ is a Banach space. We also investigate the characterizations of the space $mathcal{WB}_mu(B_X)$ of $Y$-valued, locally bounded, weakly holomorphic functions associated with the Bloch-type space $mathcal B_mu(B_X)$ of scalar-valued functions in the sense that $fin mathcal{WB}_mu(B_X)$ if $wcirc f in mathcal B_mu(B_X)$ for every $w in mathcal W,$ a separating subspace of the dual $Y'$ of $Y.$","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45854803","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 real univariate polynomials with all roots real. Such a polynomial with c sign changes and p sign preservations in the sequence of its coefficients has c positive and p negative roots counted with multiplicity. Suppose that all moduli of roots are distinct; we consider them as ordered on the positive half-axis. We ask the question: If the positions of the sign changes are known, what can the positions of the moduli of negative roots be? We prove several new results which show how far from trivial the answer to this question is.
{"title":"Beyond Descartes’ rule of signs","authors":"V. Kostov","doi":"10.33205/cma.1252639","DOIUrl":"https://doi.org/10.33205/cma.1252639","url":null,"abstract":"We consider real univariate polynomials with all roots real. Such a polynomial with c sign changes and p sign preservations in the sequence of its coefficients has c positive and p negative roots counted with multiplicity. Suppose that all moduli of roots are distinct; we consider them as ordered on the positive half-axis. We ask the question: If the positions of the sign changes are known, what can the positions of the moduli of negative roots be? We prove several new results which show how far from trivial the answer to this question is.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42598288","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":"Directs estimates and a Voronovskaja-type formula for Mihesan operators","authors":"J. Bustamante","doi":"10.33205/cma.1169884","DOIUrl":"https://doi.org/10.33205/cma.1169884","url":null,"abstract":"We present an estimate for the rate of convergence of Mihesan operators in polynomial weighted spaces. A Voronovskaja-type theorem is included.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44082330","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}
Present work contains a method to obtain Jackson and Stechkin type inequalities of approximation by integral functions of finite degree (IFFD) in some variable exponent Lebesgue space of real functions defined on $boldsymbol{R}:=left( -infty ,+infty right) $. To do this, we employ a transference theorem which produce norm inequalities starting from norm inequalities in $mathcal{C}(boldsymbol{R})$, the class of bounded uniformly continuous functions defined on $boldsymbol{R}$. Let $Bsubseteq boldsymbol{R}$ be a measurable set, $pleft( xright) :Brightarrow lbrack 1,infty )$ be a measurable function. For the class of functions $f$ belonging to variable exponent Lebesgue spaces $L_{pleft( xright) }left( Bright) $, we consider difference operator $left( I-T_{delta }right)^{r}fleft( cdot right) $ under the condition that $p(x)$ satisfies the log-Hölder continuity condition and $1leq mathop{rm ess ; inf} limitsnolimits_{xin B}p(x)$, $mathop{rm ess ; sup}limitsnolimits_{xin B}p(x)
{"title":"Exponential approximation in variable exponent Lebesgue spaces on the real line","authors":"R. Akgün","doi":"10.33205/cma.1167459","DOIUrl":"https://doi.org/10.33205/cma.1167459","url":null,"abstract":"Present work contains a method to obtain Jackson and Stechkin type inequalities of approximation by integral functions of finite degree (IFFD) in some variable exponent Lebesgue space of real functions defined on $boldsymbol{R}:=left( -infty ,+infty right) $. To do this, we employ a transference theorem which produce norm inequalities starting from norm inequalities in $mathcal{C}(boldsymbol{R})$, the class of bounded uniformly continuous functions defined on $boldsymbol{R}$. Let $Bsubseteq boldsymbol{R}$ be a measurable set, $pleft( xright) :Brightarrow lbrack 1,infty )$ be a measurable function. For the class of functions $f$ belonging to variable exponent Lebesgue spaces $L_{pleft( xright) }left( Bright) $, we consider difference operator $left( I-T_{delta }right)^{r}fleft( cdot right) $ under the condition that $p(x)$ satisfies the log-Hölder continuity condition and $1leq mathop{rm ess ; inf} limitsnolimits_{xin B}p(x)$, $mathop{rm ess ; sup}limitsnolimits_{xin B}p(x)<infty $, where $I$ is the identity operator, $rin mathrm{N}:=left{ 1,2,3,cdots right} $, $delta geq 0$ and\u0000 $$\u0000 T_{delta }fleft( xright) =frac{1}{delta }intnolimits_{0}^{delta\u0000 }fleft( x+tright) dt, xin boldsymbol{R},\u0000 T_{0}equiv I,\u0000 $$\u0000 is the forward Steklov operator. It is proved that\u0000 $$\u0000 leftVert left( I-T_{delta }right) ^{r}frightVert _{pleft( cdot\u0000 right) }\u0000 $$\u0000 is a suitable measure of smoothness for functions in $L_{pleft( xright)\u0000 }left( Bright) $, where $leftVert cdot rightVert _{pleft( cdot\u0000 right) }$ is Luxemburg norm in $L_{pleft( xright) }left( Bright) .$ We\u0000 obtain main properties of difference operator $leftVert left( I-T_{delta\u0000 }right) ^{r}frightVert _{pleft( cdot right) }$ in $L_{pleft( xright)\u0000 }left( Bright) .$ We give proof of direct and inverse theorems of\u0000 approximation by IFFD in $L_{pleft( xright) }left( boldsymbol{R}right)\u0000 . $","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45265452","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":null,"pages":null},"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}
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":null,"pages":null},"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}
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":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":"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":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":"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}
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