In a previous paper, we studied the Christoffel transforms of multiple orthogonal polynomials by means of adding a finitely supported measure to the multiple orthogonality system. This approach was able to handle the Christoffel transforms of the form $(Phimu_1,dots,Phimu_r)$ for a polynomial $Phi$, where $Phimu_j$ is the linear functional defined by $$f(x)mapsto int f(x)Phi(x)dmu_j(x).$$ For these systems we derived determinantal formulas generalizing Christoffel's classical theorem. In the current paper, we generalize these formulas to consider the case of rational perturbations $$Big(frac{Phi_1}{Psi_{1}} mu_1,dots,frac{Phi_r}{Psi_r}mu_rBig),$$ for any polynomials $Phi_1,dots,Phi_r$ and $Psi_1,dots,Psi_r$. This includes the general Christoffel transforms $(Phi_1mu_1,dots,Phi_rmu_r)$ with $r$ arbitrary polynomials {$Phi_1,dots,Phi_r$,} as well as the analogous Geronimus transforms. This generalizes a theorem of Uvarov to the multiple orthogonality setting. We allow zeros of the numerators and denominators to overlap which permits addition of pure point measure. The formulas are derived for multiple orthogonal polynomials of type I and type II for any multi-index.
在之前的一篇论文中，我们通过在多正交系统中添加有限支持度量的方法研究了多正交多项式的 Christoffel 变换。这种方法能够处理多项式$Phi$的$(Phimu_1,dots,Phimu_r)$形式的克里斯托弗尔变换，其中$Phimu_j$是由$$f(x)mapstointf(x)Phi(x)dmu_j(x)定义的线性函数。$$ 对于这些系统，我们从克里斯托弗的经典定理中总结出了行列式公式。在本文中，我们将这些公式推广到考虑有理扰动的情况$$Big(frac{Phi_1}{Psi_{1}} mu_1、dots,frac{Phi_r}{Psi_r}mu_r/Big),$$对于任何多项式$Phi_1,dots,Phi_r$和$Psi_1,dots,Psi_r$.这包括一般的 Christoffel 变换 $(Phi_1mu_1,dots,Phi_rmu_r)$与 $r$ 任意多项式 {$Phi_1,dots,Phi_r$,} 以及类似的 Geronimus 变换。这将乌瓦洛夫的一个定理推广到了多重正交性环境中。我们允许分子和分母的零点重叠，这就允许增加纯点量。推导出了任意多指数的 I 型和 II 型多重正交多项式的公式。
{"title":"Determinantal Formulas for Rational Perturbations of Multiple Orthogonality Measures","authors":"Rostyslav Kozhan, Marcus Vaktnäs","doi":"arxiv-2407.13961","DOIUrl":"https://doi.org/arxiv-2407.13961","url":null,"abstract":"In a previous paper, we studied the Christoffel transforms of multiple\u0000orthogonal polynomials by means of adding a finitely supported measure to the\u0000multiple orthogonality system. This approach was able to handle the Christoffel\u0000transforms of the form $(Phimu_1,dots,Phimu_r)$ for a polynomial $Phi$,\u0000where $Phimu_j$ is the linear functional defined by $$f(x)mapsto int\u0000f(x)Phi(x)dmu_j(x).$$ For these systems we derived determinantal formulas\u0000generalizing Christoffel's classical theorem. In the current paper, we\u0000generalize these formulas to consider the case of rational perturbations\u0000$$Big(frac{Phi_1}{Psi_{1}} mu_1,dots,frac{Phi_r}{Psi_r}mu_rBig),$$\u0000for any polynomials $Phi_1,dots,Phi_r$ and $Psi_1,dots,Psi_r$. This\u0000includes the general Christoffel transforms $(Phi_1mu_1,dots,Phi_rmu_r)$\u0000with $r$ arbitrary polynomials {$Phi_1,dots,Phi_r$,} as well as the\u0000analogous Geronimus transforms. This generalizes a theorem of Uvarov to the\u0000multiple orthogonality setting. We allow zeros of the numerators and\u0000denominators to overlap which permits addition of pure point measure. The\u0000formulas are derived for multiple orthogonal polynomials of type I and type II\u0000for any multi-index.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141746377","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 main goal of this note is to characterize the necessary and sufficient conditions for a composition operator to act between spaces of mappings of bounded Wiener variation in a normed-valued setting. The necessary and sufficient conditions for local boundedness of such operators are also discussed.
{"title":"Josephy's theorem, revisited","authors":"Daria Bugajewska, Piotr Kasprzak","doi":"arxiv-2407.14169","DOIUrl":"https://doi.org/arxiv-2407.14169","url":null,"abstract":"The main goal of this note is to characterize the necessary and sufficient\u0000conditions for a composition operator to act between spaces of mappings of\u0000bounded Wiener variation in a normed-valued setting. The necessary and\u0000sufficient conditions for local boundedness of such operators are also\u0000discussed.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"93 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745019","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 real and complex zeros of the parabolic cylinder function $U(a,z)$ are studied. Asymptotic expansions for the zeros are derived, involving the zeros of Airy functions, and these are valid for $a$ positive or negative and large in absolute value, uniformly for unbounded $z$ (real or complex). The accuracy of the approximations of the complex zeros is then demonstrated with some comparative tests using a highly precise numerical algorithm for finding the complex zeros of the function.
{"title":"Uniform asymptotic expansions for the zeros of parabolic cylinder functions","authors":"T. M. Dunster, A. Gil, D. Ruiz-Antolin, J. Segura","doi":"arxiv-2407.13936","DOIUrl":"https://doi.org/arxiv-2407.13936","url":null,"abstract":"The real and complex zeros of the parabolic cylinder function $U(a,z)$ are\u0000studied. Asymptotic expansions for the zeros are derived, involving the zeros\u0000of Airy functions, and these are valid for $a$ positive or negative and large\u0000in absolute value, uniformly for unbounded $z$ (real or complex). The accuracy\u0000of the approximations of the complex zeros is then demonstrated with some\u0000comparative tests using a highly precise numerical algorithm for finding the\u0000complex zeros of the function.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"36 Suppl 2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744987","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 make use of the Laplace transform and gamma function to construct a new integral transform having the property of mapping a derivative to the backward difference, whence we derive a method for solving difference equations and, relying on classical orthogonal polynomials, for obtaining combinatorial identities. A table of some basic functions is given in the Appendix.
{"title":"On the gamma transform and its applications","authors":"Slobodan B. Tričković, Miomir S. Stanković","doi":"arxiv-2407.13812","DOIUrl":"https://doi.org/arxiv-2407.13812","url":null,"abstract":"We make use of the Laplace transform and gamma function to construct a new\u0000integral transform having the property of mapping a derivative to the backward\u0000difference, whence we derive a method for solving difference equations and,\u0000relying on classical orthogonal polynomials, for obtaining combinatorial\u0000identities. A table of some basic functions is given in the Appendix.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"126 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744989","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}
One of the most celebrated problems in Euclidean Harmonic analysis is the Carleson's problem: determining the optimal regularity of the initial condition $f$ of the Schr"odinger equation given by begin{equation*}begin{cases} ifrac{partial u}{partial t} =Delta u:,: (x,t) in mathbb{R}^n times mathbb{R} u(0,cdot)=f:, text{ on } mathbb{R}^n :, end{cases}end{equation*} in terms of the index $alpha$ such that $f$ belongs to the inhomogeneous Sobolev space $H^alpha(mathbb{R}^n)$ , so that the solution of the Schr"odinger operator $u$ converges pointwise to $f$, $lim_{t to 0+} u(x,t)=f(x)$, almost everywhere. In this article, we consider the Carleson's problem for the Schr"odinger equation with radial initial data on Damek-Ricci spaces and obtain the sharp bound up to the endpoint $alpha ge 1/4$, which agrees with the classical Euclidean case.
欧几里得谐波分析中最著名的问题之一是卡莱森问题：确定薛定谔方程的初始条件f的最优正则性 给定方程为 u(0,cdot)=fend{cases}end{equation*} in terms of the index $alpha$ such that $f$ belongsto the inhomogeneous Sobolev space $H^alpha(mathbb{R}^n)$ , so that thesolution of the Schr"odinger operator $u$ converges pointwise to $f$, $lim_{tto 0+} u(x,t)=f(x)$, almost everywhere.在本文中，我们考虑了在达梅克-里奇空间上具有径向初始数据的薛定谔方程的卡勒森问题，并得到了直到端点 $alpha ge1/4$ 的尖锐约束，这与经典欧几里得情况一致。
{"title":"Regularity and pointwise convergence of solutions of the Schrödinger operator with radial initial data on Damek-Ricci spaces","authors":"Utsav Dewan","doi":"arxiv-2407.13736","DOIUrl":"https://doi.org/arxiv-2407.13736","url":null,"abstract":"One of the most celebrated problems in Euclidean Harmonic analysis is the\u0000Carleson's problem: determining the optimal regularity of the initial condition\u0000$f$ of the Schr\"odinger equation given by begin{equation*}begin{cases}\u0000ifrac{partial u}{partial t} =Delta u:,: (x,t) in mathbb{R}^n times\u0000mathbb{R} u(0,cdot)=f:, text{ on } mathbb{R}^n :,\u0000end{cases}end{equation*} in terms of the index $alpha$ such that $f$ belongs\u0000to the inhomogeneous Sobolev space $H^alpha(mathbb{R}^n)$ , so that the\u0000solution of the Schr\"odinger operator $u$ converges pointwise to $f$, $lim_{t\u0000to 0+} u(x,t)=f(x)$, almost everywhere. In this article, we consider the\u0000Carleson's problem for the Schr\"odinger equation with radial initial data on\u0000Damek-Ricci spaces and obtain the sharp bound up to the endpoint $alpha ge\u00001/4$, which agrees with the classical Euclidean case.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141746460","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 classical Borsuk's non-retract theorem asserts that a unit sphere in $mathbb{R}^n$ is not a continuous retract of the unit closed ball. We will show that such a unit sphere is a piecewise continuous retract of the unit closed ball.
{"title":"On Borsuk's non-retract theorem","authors":"Waldemar Sieg","doi":"arxiv-2407.13395","DOIUrl":"https://doi.org/arxiv-2407.13395","url":null,"abstract":"The classical Borsuk's non-retract theorem asserts that a unit sphere in\u0000$mathbb{R}^n$ is not a continuous retract of the unit closed ball. We will\u0000show that such a unit sphere is a piecewise continuous retract of the unit\u0000closed ball.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745017","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 identify a connection between the Christoffel transform of orthogonal polynomials and multiple orthogonality systems containing a finitely supported measure. In consequence, the compatibility relations for the nearest neighbour recurrence coefficients provide a new algorithm for the computation of the Jacobi coefficients of the one-step or multi-step Christoffel transforms. More generally, we investigate multiple orthogonal polynomials associated with the system of measures obtained by applying a Christoffel transform to each of the orthogonality measures. We present an algorithm for computing the transformed recurrence coefficients, and determinantal formulas for the transformed multiple orthogonal polynomials of type I and type II. Finally, we show that zeros of multiple orthogonal polynomials of an Angelesco or an AT system interlace with the zeros of the polynomial corresponding to the one-step Christoffel transform. This allows us to prove a number of interlacing properties satisfied by the multiple orthogonality analogues of classical orthogonal polynomials. For the discrete polynomials, this also produces an estimate on the smallest distance between consecutive zeros.
我们发现了正交多项式的 Christoffel 变换与包含有限支持度量的多重正交系统之间的联系。因此，近邻复现系数的相容关系为计算一步或多步克里斯托弗变换的雅可比系数提供了一种新算法。更一般地说，我们研究了与通过对每个正交度量应用 Christoffel 变换而得到的度量系统相关的多个正交多项式。我们提出了一种计算变换后复现系数的算法，以及 I 型和 II 型变换后多重正交多项式的行列式。最后，我们证明了安立斯科或 AT 系统的多重正交多项式的零点与一步克里斯托弗变换对应的多项式的零点交错。这样，我们就可以证明经典正交多项式的多重正交性类似物所满足的一系列交错特性。对于离散多项式来说，这也产生了连续零点之间最小距离的估计值。
{"title":"Christoffel Transform and Multiple Orthogonal Polynomials","authors":"Rostyslav Kozhan, Marcus Vaktnäs","doi":"arxiv-2407.13946","DOIUrl":"https://doi.org/arxiv-2407.13946","url":null,"abstract":"We identify a connection between the Christoffel transform of orthogonal\u0000polynomials and multiple orthogonality systems containing a finitely supported\u0000measure. In consequence, the compatibility relations for the nearest neighbour\u0000recurrence coefficients provide a new algorithm for the computation of the\u0000Jacobi coefficients of the one-step or multi-step Christoffel transforms. More\u0000generally, we investigate multiple orthogonal polynomials associated with the\u0000system of measures obtained by applying a Christoffel transform to each of the\u0000orthogonality measures. We present an algorithm for computing the transformed\u0000recurrence coefficients, and determinantal formulas for the transformed\u0000multiple orthogonal polynomials of type I and type II. Finally, we show that\u0000zeros of multiple orthogonal polynomials of an Angelesco or an AT system\u0000interlace with the zeros of the polynomial corresponding to the one-step\u0000Christoffel transform. This allows us to prove a number of interlacing\u0000properties satisfied by the multiple orthogonality analogues of classical\u0000orthogonal polynomials. For the discrete polynomials, this also produces an\u0000estimate on the smallest distance between consecutive zeros.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745014","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 have derived alternative closed-form formulas for the trigonometric series over sine or cosine functions when the immediate replacement of the parameter appearing in the denominator with a positive integer gives rise to a singularity. By applying the Choi-Srivastava theorem, we reduce these trigonometric series to expressions over Hurwitz's zeta function derivative.
{"title":"On closed forms of some trigonometric series","authors":"Slobodan B. Tričković, Miomir S. Stanković","doi":"arxiv-2407.12885","DOIUrl":"https://doi.org/arxiv-2407.12885","url":null,"abstract":"We have derived alternative closed-form formulas for the trigonometric series\u0000over sine or cosine functions when the immediate replacement of the parameter\u0000appearing in the denominator with a positive integer gives rise to a\u0000singularity. By applying the Choi-Srivastava theorem, we reduce these\u0000trigonometric series to expressions over Hurwitz's zeta function derivative.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745015","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 conjugate equation driven by two families of finite maps on the unit interval satisfying a compatibility condition. This framework contains de Rham's functional equations. We consider some real analytic properties of the solution in the case that the equation is driven by non-affine maps, in particular, linear fractional transformations. We give sufficient conditions for the regularity in the sense of Ullman-Stahl-Totik and for the singularity of the solution.
{"title":"Regularity and singularity for conjugate equations driven by linear fractional transformations","authors":"Kazuki Okamura","doi":"arxiv-2407.11565","DOIUrl":"https://doi.org/arxiv-2407.11565","url":null,"abstract":"We consider the conjugate equation driven by two families of finite maps on\u0000the unit interval satisfying a compatibility condition. This framework contains\u0000de Rham's functional equations. We consider some real analytic properties of\u0000the solution in the case that the equation is driven by non-affine maps, in\u0000particular, linear fractional transformations. We give sufficient conditions\u0000for the regularity in the sense of Ullman-Stahl-Totik and for the singularity\u0000of the solution.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720017","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 analyze the Bass and SI models for the spreading of innovations and epidemics, respectively, on homogeneous complete networks, circular networks, and heterogeneous complete networks with two homogeneous groups. We allow the network parameters to be time dependent, which is a prerequisite for the analysis of optimal strategies on networks. Using a novel top-down analysis of the master equations, we present a simple proof for the monotone convergence of these models to their respective infinite-population limits. This leads to explicit expressions for the expected adoption or infection level in the Bass and SI models, respectively, on infinite homogeneous complete and circular networks, and on heterogeneous complete networks with two homogeneous groups with time-dependent parameters.
我们分别在同质完整网络、环形网络和有两个同质群体的异质完整网络上分析了创新和流行病传播的巴斯模型和SI模型。我们允许网络参数与时间相关，这是分析网络最优策略的前提条件。通过对主方程进行新颖的自顶向下分析，我们给出了这些模型单调收敛到各自无限人口极限的简单证明。由此，我们分别得出了在无限同质完整网络和循环网络上，以及在具有两个同质组且参数随时间变化的异质完整网络上，Bass 和 SI 模型的预期采用或感染水平的明确表达式。
{"title":"Monotone convergence of spreading processes on networks","authors":"Gadi Fibich, Amit Golan, Steven Schochet","doi":"arxiv-2407.10816","DOIUrl":"https://doi.org/arxiv-2407.10816","url":null,"abstract":"We analyze the Bass and SI models for the spreading of innovations and\u0000epidemics, respectively, on homogeneous complete networks, circular networks,\u0000and heterogeneous complete networks with two homogeneous groups. We allow the\u0000network parameters to be time dependent, which is a prerequisite for the\u0000analysis of optimal strategies on networks. Using a novel top-down analysis of\u0000the master equations, we present a simple proof for the monotone convergence of\u0000these models to their respective infinite-population limits. This leads to\u0000explicit expressions for the expected adoption or infection level in the Bass\u0000and SI models, respectively, on infinite homogeneous complete and circular\u0000networks, and on heterogeneous complete networks with two homogeneous groups\u0000with time-dependent parameters.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720019","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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