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Determinantal Formulas for Rational Perturbations of Multiple Orthogonality Measures 多重正交度量有理扰动的确定性公式
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-19 DOI: arxiv-2407.13961
Rostyslav Kozhan, Marcus Vaktnäs
{"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}
引用次数: 0
Josephy's theorem, revisited 约瑟夫定理重温
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-19 DOI: arxiv-2407.14169
Daria Bugajewska, Piotr Kasprzak
{"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}
引用次数: 0
Uniform asymptotic expansions for the zeros of parabolic cylinder functions 抛物柱面函数零点的统一渐近展开式
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-18 DOI: arxiv-2407.13936
T. M. Dunster, A. Gil, D. Ruiz-Antolin, J. Segura
{"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}
引用次数: 0
On the gamma transform and its applications 关于伽马变换及其应用
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-18 DOI: arxiv-2407.13812
Slobodan B. Tričković, Miomir S. Stanković
{"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}
引用次数: 0
Regularity and pointwise convergence of solutions of the Schrödinger operator with radial initial data on Damek-Ricci spaces 达梅克-里奇空间上具有径向初始数据的薛定谔算子解的正则性和点收敛性
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-18 DOI: arxiv-2407.13736
Utsav Dewan
{"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}
引用次数: 0
On Borsuk's non-retract theorem 关于博尔苏克的非撤回定理
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-18 DOI: arxiv-2407.13395
Waldemar Sieg
{"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}
引用次数: 0
Christoffel Transform and Multiple Orthogonal Polynomials 克里斯托弗变换和多重正交多项式
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-18 DOI: arxiv-2407.13946
Rostyslav Kozhan, Marcus Vaktnäs
{"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}
引用次数: 0
On closed forms of some trigonometric series 关于一些三角函数级数的封闭形式
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-16 DOI: arxiv-2407.12885
Slobodan B. Tričković, Miomir S. Stanković
{"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}
引用次数: 0
Regularity and singularity for conjugate equations driven by linear fractional transformations 线性分数变换驱动的共轭方程的正则性和奇异性
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-16 DOI: arxiv-2407.11565
Kazuki Okamura
{"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}
引用次数: 0
Monotone convergence of spreading processes on networks 网络传播过程的单调收敛性
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-15 DOI: arxiv-2407.10816
Gadi Fibich, Amit Golan, Steven Schochet
{"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}
引用次数: 0
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