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Multiple Laguerre polynomials: Combinatorial model and Stieltjes moment representation 多重拉盖尔多项式:组合模型与Stieltjes矩表示
arXiv: Classical Analysis and ODEs Pub Date : 2021-04-17 DOI: 10.1090/proc/15775
A. Sokal
{"title":"Multiple Laguerre polynomials: Combinatorial model and Stieltjes moment representation","authors":"A. Sokal","doi":"10.1090/proc/15775","DOIUrl":"https://doi.org/10.1090/proc/15775","url":null,"abstract":"I give a combinatorial interpretation of the multiple Laguerre polynomials of the first kind of type II, generalizing the digraph model found by Foata and Strehl for the ordinary Laguerre polynomials. I also give an explicit integral representation for these polynomials, which shows that they form a multidimensional Stieltjes moment sequence whenever $x le 0$.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82927320","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: 4
Stability and measurability of the modified lower dimension 修正下维的稳定性和可测性
arXiv: Classical Analysis and ODEs Pub Date : 2021-02-25 DOI: 10.1090/proc/16029
R. Balka, Márton Elekes, V. Kiss
{"title":"Stability and measurability of the modified lower dimension","authors":"R. Balka, Márton Elekes, V. Kiss","doi":"10.1090/proc/16029","DOIUrl":"https://doi.org/10.1090/proc/16029","url":null,"abstract":"The lower dimension $dim_L$ is the dual concept of the Assouad dimension. As it fails to be monotonic, Fraser and Yu introduced the modified lower dimension $dim_{ML}$ by making the lower dimension monotonic with the simple formula $dim_{ML} X=sup{dim_L E: Esubset X}$. \u0000As our first result we prove that the modified lower dimension is finitely stable in any metric space, answering a question of Fraser and Yu. \u0000We prove a new, simple characterization for the modified lower dimension. For a metric space $X$ let $mathcal{K}(X)$ denote the metric space of the non-empty compact subsets of $X$ endowed with the Hausdorff metric. As an application of our characterization, we show that the map $dim_{ML} colon mathcal{K}(X)to [0,infty]$ is Borel measurable. More precisely, it is of Baire class $2$, but in general not of Baire class $1$. This answers another question of Fraser and Yu. \u0000Finally, we prove that the modified lower dimension is not Borel measurable defined on the closed sets of $ell^1$ endowed with the Effros Borel structure.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76145858","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
Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle 一维和高维正则测度的加性能量，以及分形测不准原理
arXiv: Classical Analysis and ODEs Pub Date : 2020-12-04 DOI: 10.15781/GW9Q-K252
Laura Cladek, T. Tao
{"title":"Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle","authors":"Laura Cladek, T. Tao","doi":"10.15781/GW9Q-K252","DOIUrl":"https://doi.org/10.15781/GW9Q-K252","url":null,"abstract":"We obtain new bounds on the additive energy of (Ahlfors-David type) regular measures in both one and higher dimensions, which implies expansion results for sums and products of the associated regular sets, as well as more general nonlinear functions of these sets. As a corollary of the higher-dimensional results we obtain some new cases of the fractal uncertainty principle in odd dimensions.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74487574","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: 11
Roots of Gårding hyperbolic polynomials 格尔丁双曲多项式的根
arXiv: Classical Analysis and ODEs Pub Date : 2020-12-02 DOI: 10.1090/PROC/15634
A. Rainer
{"title":"Roots of Gårding hyperbolic polynomials","authors":"A. Rainer","doi":"10.1090/PROC/15634","DOIUrl":"https://doi.org/10.1090/PROC/15634","url":null,"abstract":"We explore the regularity of the roots of Garding hyperbolic polynomials and real stable polynomials. As an application we obtain new regularity results of Sobolev type for the eigenvalues of Hermitian matrices and for the singular values of arbitrary matrices. These results are optimal among all Sobolev spaces.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90257698","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
Simpson’s Rule Revisited 辛普森法则重述
arXiv: Classical Analysis and ODEs Pub Date : 2020-11-27 DOI: 10.1142/s0219876221500110
S. Simić, B. Bin-Mohsin
{"title":"Simpson’s Rule Revisited","authors":"S. Simić, B. Bin-Mohsin","doi":"10.1142/s0219876221500110","DOIUrl":"https://doi.org/10.1142/s0219876221500110","url":null,"abstract":"In this article we give some refinements of Simpson's Rule in cases when it is not applicable in it's classical form i.e., when the target function is not four times differentiable on a given interval. Some sharp two-sided inequalities for an extended form of Simpson's Rule are also proven.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80773661","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
Extrapolation for multilinear compact operators and applications 多线性紧算子的外推及其应用
arXiv: Classical Analysis and ODEs Pub Date : 2020-11-26 DOI: 10.1090/tran/8645
Mingming Cao, Andrea Olivo, K. Yabuta
{"title":"Extrapolation for multilinear compact operators and applications","authors":"Mingming Cao, Andrea Olivo, K. Yabuta","doi":"10.1090/tran/8645","DOIUrl":"https://doi.org/10.1090/tran/8645","url":null,"abstract":"This paper is devoted to studying the Rubio de Francia extrapolation for multilinear compact operators. It allows one to extrapolate the compactness of $T$ from just one space to the full range of weighted spaces, whenever an $m$-linear operator $T$ is bounded on weighted Lebesgue spaces. This result is indeed established in terms of the multilinear Muckenhoupt weights $A_{vec{p}, vec{r}}$, and the limited range of the $L^p$ scale. As applications, we obtain the weighted compactness of commutators of many multilinear operators, including multilinear $omega$-Calder'{o}n-Zygmund operators, multilinear Fourier multipliers, bilinear rough singular integrals and bilinear Bochner-Riesz means.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88853972","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: 20
Decaying positive global solutions of second order difference equations with mean curvature operator 具有平均曲率算子的二阶差分方程的衰减正全局解
arXiv: Classical Analysis and ODEs Pub Date : 2020-11-24 DOI: 10.14232/EJQTDE.2020.1.72
Z. Došlá, S. Matucci, P. Řehák
{"title":"Decaying positive global solutions of second order difference equations with mean curvature operator","authors":"Z. Došlá, S. Matucci, P. Řehák","doi":"10.14232/EJQTDE.2020.1.72","DOIUrl":"https://doi.org/10.14232/EJQTDE.2020.1.72","url":null,"abstract":"A boundary value problem on an unbounded domain, associated to difference equations with the Euclidean mean curvature operator is considered. The existence of solutions which are positive on the whole domain and decaying at infinity is examined by proving new Sturm comparison theorems for linear difference equations and using a fixed point approach based on a linearization device. %The process from the continuous problem to discrete one is examined, too. The process of discretization of the boundary value problem on the unbounded domain is examined, and some discrepancies between the discrete and the continuous case are pointed out, too.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88002286","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: 2
Finite-part integration in the presence of competing singularities: Transformation equations for the hypergeometric functions arising from finite-part integration 竞争奇点下的有限部分积分:由有限部分积分引起的超几何函数的变换方程
arXiv: Classical Analysis and ODEs Pub Date : 2020-11-20 DOI: 10.1063/5.0038274
L. Villanueva, E. Galapon
{"title":"Finite-part integration in the presence of competing singularities: Transformation equations for the hypergeometric functions arising from finite-part integration","authors":"L. Villanueva, E. Galapon","doi":"10.1063/5.0038274","DOIUrl":"https://doi.org/10.1063/5.0038274","url":null,"abstract":"Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite part of divergent integrals [E.A. Galapon, {it Proc. R. Soc. A 473, 20160567} (2017)]. Current application of the method involves exact and asymptotic evaluation of the generalized Stieltjes transform $int_0^a f(x)/(omega + x)^{rho} , mathrm{d}x$ under the assumption that the extension of $f(x)$ in the complex plane is entire. In this paper, the method is elaborated further and extended to accommodate the presence of competing singularities of the complex extension of $f(x)$. Finite part integration is then applied to derive consequences of known Stieltjes integral representations of the Gauss function and the generalized hypergeometric function which involve Stieltjes transforms of functions with complex extensions having singularities in the complex plane. Transformation equations for the Gauss function are obtained from which known transformation equations are shown to follow. Also, building on the results for the Gauss function, transformation equations involving the generalized hypergeometric function $,_3F_2$ are derived.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90206229","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: 5
Orthogonality of the Dickson polynomials of the $(k+1)$-th kind 第(k+1)类的Dickson多项式的正交性
arXiv: Classical Analysis and ODEs Pub Date : 2020-11-20 DOI: 10.33044/REVUMA.V62N1A01
D. Dominici
{"title":"Orthogonality of the Dickson polynomials of the $(k+1)$-th kind","authors":"D. Dominici","doi":"10.33044/REVUMA.V62N1A01","DOIUrl":"https://doi.org/10.33044/REVUMA.V62N1A01","url":null,"abstract":"We study the Dickson polynomials of the (k+1)-th kind over the field of complex numbers. We show that they are a family of co-recursive orthogonal polynomials with respect to a quasi-definite moment functional L_{k}. We find an integral representation for L_{k} and compute explicit expressions for all of its moments.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88506840","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: 3
Applications of a duality between generalized trigonometric and hyperbolic functions. 广义三角函数与双曲函数对偶的应用。
arXiv: Classical Analysis and ODEs Pub Date : 2020-11-13 DOI: 10.1016/j.jmaa.2021.12524
Hiroki Miyakawa, S. Takeuchi
{"title":"Applications of a duality between generalized trigonometric and hyperbolic functions.","authors":"Hiroki Miyakawa, S. Takeuchi","doi":"10.1016/j.jmaa.2021.12524","DOIUrl":"https://doi.org/10.1016/j.jmaa.2021.12524","url":null,"abstract":"","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81976678","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: 2
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