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 $x le 0$.
本文将Foata和Strehl对普通拉盖尔多项式的有向图模型进行了推广，给出了第一类II型多重拉盖尔多项式的组合解释。我还给出了这些多项式的显式积分表示，表明它们在x le 0$时形成多维Stieltjes矩序列。
{"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}
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 $dim_{ML} X=sup{dim_L E: Esubset X}$. As our first result we prove that the modified lower dimension is finitely stable in any metric space, answering a question of Fraser and Yu. We 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. Finally, 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.
{"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}
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.
{"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}
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.
{"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}
Pub Date : 2020-11-27DOI: 10.1142/s0219876221500110
S. Simić, B. Bin-Mohsin
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.
{"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}
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.
本文研究了多线性紧算子的Rubio de Francia外推。当一个$m$ -线性算子$T$在加权Lebesgue空间上有界时，它允许我们从一个空间向整个加权空间外推$T$的紧性。这个结果确实是建立在多元线性Muckenhoupt权重$A_{vec{p}, vec{r}}$和$L^p$量表的有限范围内。作为应用，我们得到了许多多重线性算子的交换子的加权紧性，包括多重线性$omega$ -Calderón-Zygmund算子、多重线性傅立叶乘子、双线性粗糙奇异积分和双线性Bochner-Riesz均值。
{"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}
Pub Date : 2020-11-24DOI: 10.14232/EJQTDE.2020.1.72
Z. Došlá, S. Matucci, P. Řehák
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.
{"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}
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.
{"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}
Pub Date : 2020-11-20DOI: 10.33044/REVUMA.V62N1A01
D. Dominici
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.
{"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}
Pub Date : 2020-11-13DOI: 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}
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