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A molecular decomposition for $H^p(mathbb{Z}^n)$ and applications $H^p(mathbb{Z}^n)$ 的分子分解及其应用
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-08-18 DOI: arxiv-2408.09528
Pablo Rocha
{"title":"A molecular decomposition for $H^p(mathbb{Z}^n)$ and applications","authors":"Pablo Rocha","doi":"arxiv-2408.09528","DOIUrl":"https://doi.org/arxiv-2408.09528","url":null,"abstract":"In this work, for the range $frac{n-1}{n} < p leq 1$, we give a molecular\u0000reconstruction theorem for $H^p(mathbb{Z}^n)$. As an application of this\u0000result and the atomic decomposition developed by S. Boza and M. Carro in [Proc.\u0000R. Soc. Edinb., 132 A (1) (2002), 25-43], we prove that the discrete Riesz\u0000potential $I_{alpha}$ defined on $mathbb{Z}^n$ is a bounded operator\u0000$H^p(mathbb{Z}^n) to H^q(mathbb{Z}^n)$ for $frac{n-1}{n} < p <\u0000frac{n}{alpha}$ and $frac{1}{q} = frac{1}{p} - frac{alpha}{n}$, where $0\u0000< alpha < n$.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209763","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
Stability of the concentration inequality on polynomials 多项式集中不等式的稳定性
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-08-14 DOI: arxiv-2408.07424
María Ángeles García-Ferrero, Joaquim Ortega-Cerdà
{"title":"Stability of the concentration inequality on polynomials","authors":"María Ángeles García-Ferrero, Joaquim Ortega-Cerdà","doi":"arxiv-2408.07424","DOIUrl":"https://doi.org/arxiv-2408.07424","url":null,"abstract":"In this paper, we study the stability of the concentration inequality for\u0000one-dimensional complex polynomials. We provide the stability of the local\u0000concentration inequality and a global version using a Wehrl-type entropy.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209765","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
The finite bivariate biorthogonal I -- Konhauser polynomials 有限双变量双谐波 I - 康豪斯多项式
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-08-14 DOI: arxiv-2408.07811
Esra Güldoğan Lekesiz, Bayram Çekim, Mehmet Ali Özarslan
{"title":"The finite bivariate biorthogonal I -- Konhauser polynomials","authors":"Esra Güldoğan Lekesiz, Bayram Çekim, Mehmet Ali Özarslan","doi":"arxiv-2408.07811","DOIUrl":"https://doi.org/arxiv-2408.07811","url":null,"abstract":"In this paper, a finite set of biorthogonal polynomials in two variables is\u0000produced using Konhauser polynomials. Some properties containing operational\u0000and integral representation, Laplace transform, fractional calculus operators\u0000of this family are studied. Also, computing Fourier transform for the new set,\u0000a new family of biorthogonal functions are derived via Parseval's identity. On\u0000the other hand, this finite set is modified by adding two new parameters in\u0000order to have semigroup property and construct fractional calculus operators.\u0000Further, integral equation and integral operator are also derived for the\u0000modified version.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209770","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
Generalized square function estimates for curves and their conical extensions 曲线及其圆锥延伸的广义平方函数估计值
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-08-14 DOI: arxiv-2408.07248
Robert Schippa
{"title":"Generalized square function estimates for curves and their conical extensions","authors":"Robert Schippa","doi":"arxiv-2408.07248","DOIUrl":"https://doi.org/arxiv-2408.07248","url":null,"abstract":"We show sharp square function estimates for curves in the plane whose\u0000curvature degenerates at a point and estimates sharp up to endpoints for cones\u0000over these curves. To this end, for curves of finite type we extend the\u0000classical C'ordoba--Fefferman biorthogonality. For cones over degenerate\u0000curves, we analyze wave envelope estimates proved via High-Low-decomposition.\u0000The arguments are subsequently extended to the cone over the complex parabola.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209766","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
Comparison of Gini means with fixed number of variables 固定变量数量下的基尼系数比较
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-08-14 DOI: arxiv-2408.07658
Richárd Grünwald, Zsolt Páles
{"title":"Comparison of Gini means with fixed number of variables","authors":"Richárd Grünwald, Zsolt Páles","doi":"arxiv-2408.07658","DOIUrl":"https://doi.org/arxiv-2408.07658","url":null,"abstract":"In this paper, we consider the global comparison problem of Gini means with\u0000fixed number of variables on a subinterval $I$ of $mathbb{R}_+$, i.e., the\u0000following inequality begin{align}tag{$star$}label{ggcabs} G_{r,s}^{[n]}(x_1,dots,x_n) leq G_{p,q}^{[n]}(x_1,dots,x_n), end{align} where $ninmathbb{N},ngeq2$ is fixed, $(p,q),(r,s)inmathbb{R}^2$ and\u0000$x_1,dots,x_nin I$. Given a nonempty subinterval $I$ of $mathbb{R}_+$ and $ninmathbb{N}$, we\u0000introduce the relations [ Gamma_n(I):={((r,s),(p,q))inmathbb{R}^2timesmathbb{R}^2mid\u0000eqref{ggcabs}mbox{ holds for all } x_1,dots,x_nin I},qquad Gamma_infty(I):=bigcap_{n=1}^inftyGamma_n(I). ] In the paper, we investigate the properties of these sets and their\u0000dependence on $n$ and on the interval $I$ and we establish a characterizations\u0000of these sets via a constrained minimum problem by using a variant of the\u0000Lagrange multiplier rule. We also formulate two open problems at the end of the\u0000paper.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209764","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 a criterion of uniform distribution 关于均匀分布的标准
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-08-13 DOI: arxiv-2408.07061
Grigori Karagulyan, Iren Petrosyan
{"title":"On a criterion of uniform distribution","authors":"Grigori Karagulyan, Iren Petrosyan","doi":"arxiv-2408.07061","DOIUrl":"https://doi.org/arxiv-2408.07061","url":null,"abstract":"We give an extension of a criterion of van der Corput on uniform distribution\u0000of sequences. Namely, we prove that a sequence $x_n$ is uniformly distributed\u0000modulo 1 if it is weakly monotonic and satisfies the conditions $Delta^2x_nto\u00000,quad n^2Delta^2x_nto infty $. Our proof is straightforward and uses a\u0000Diophantine approximation by rational numbers, while van der Corput's approach\u0000is based on some estimates of exponential sums.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209767","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
Periodic Source Detection in Discrete Dynamical Systems via space-time sampling 通过时空采样进行离散动态系统中的周期源检测
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-08-13 DOI: arxiv-2408.06934
Akram Aldroubi, Carlos Cabrelli, Ursula Molter
{"title":"Periodic Source Detection in Discrete Dynamical Systems via space-time sampling","authors":"Akram Aldroubi, Carlos Cabrelli, Ursula Molter","doi":"arxiv-2408.06934","DOIUrl":"https://doi.org/arxiv-2408.06934","url":null,"abstract":"In this paper, we examine a discrete dynamical system defined by x(n+1) =\u0000Ax(n)+ w(n), where x takes values in a Hilbert space H and w is a periodic\u0000source with values in a fixed closed subspace W of H. Our goal is to identify\u0000conditions on some spatial sampling system G = {gj: j in J} of H that enable\u0000stable recovery of the unknown source term w from space-time samples\u0000{<x(n),g_j>: n >=0,j in J}. We provide necessary and sufficient conditions on G\u0000= {g_j }_{j in J} to ensure stable recovery of any w in W . Additionally, we\u0000explicitly construct an operator R, dependent on G, such that\u0000R{<x(n),g_j>}_n,j} = w.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209889","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
Nonlinear non-periodic homogenization: Existence, local uniqueness and estimates 非线性非周期性均质化：存在性、局部唯一性和估计值
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-08-13 DOI: arxiv-2408.06705
Lutz Recke
{"title":"Nonlinear non-periodic homogenization: Existence, local uniqueness and estimates","authors":"Lutz Recke","doi":"arxiv-2408.06705","DOIUrl":"https://doi.org/arxiv-2408.06705","url":null,"abstract":"We consider periodic homogenization with localized defects of boundary value\u0000problems for semilinear ODE systems of the type $$\u0000Big((A(x/varepsilon)+B(x/varepsilon))u'(x)+c(x,u(x))Big)'= d(x,u(x)) mbox{\u0000for } x in (0,1),; u(0)=u(1)=0. $$ Our assumptions are, roughly speaking, as\u0000follows: $A in L^infty(mathbb{R};mathbb{M}_n)$ is 1-periodic, $B in\u0000L^infty(mathbb{R};mathbb{M}_n))cap L^1(mathbb{R};mathbb{M}_n))$, $A(y)$\u0000and $A(y)+B(y)$ are positive definite uniformly with respect to $y$,\u0000$c(x,cdot),d(x,cdot)in C^1(mathbb{R}^n;mathbb{R}^n))$, $c(cdot,u) in\u0000C([0,1];mathbb{R}^n)$ and $d(cdot,u) in L^infty((0,1);mathbb{R}^n)$. For\u0000small $varepsilon>0$ we show existence of weak solutions $u=u_varepsilon$ as\u0000well as their local uniqueness for $|u-u_0|_infty approx 0$, where $u_0$ is\u0000a given non-degenerate solution to the homogenized problem, and we prove that\u0000$|u_varepsilon-u_0|_inftyto 0$ and, if $c(cdot,u)$ is $C^1$-smooth, that\u0000$|u_varepsilon-u_0|_infty=O(varepsilon)$ for $varepsilon to 0$. The main\u0000tool of the proofs is an abstract result of implicit function theorem type\u0000which in the past has been applied to singular perturbation as well as to\u0000periodic homogenization of nonlinear ODEs and PDEs and, hence, which permits a\u0000common approach to existence and local uniqueness results for singularly\u0000perturbed problems and for homogenization problems.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209768","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
A note on the problem of straight-line interpolation by ridge functions 关于脊函数直线插值问题的说明
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-08-12 DOI: arxiv-2408.06443
Azer Akhmedov, Vugar Ismailov
{"title":"A note on the problem of straight-line interpolation by ridge functions","authors":"Azer Akhmedov, Vugar Ismailov","doi":"arxiv-2408.06443","DOIUrl":"https://doi.org/arxiv-2408.06443","url":null,"abstract":"In this paper we discuss the problem of interpolation on straight lines by\u0000linear combinations of ridge functions with fixed directions. By using some\u0000geometry and/or systems of linear equations, we constructively prove that it is\u0000impossible to interpolate arbitrary data on any three or more straight lines by\u0000sums of ridge functions with two fixed directions. The general case with more\u0000straight lines and more directions is reduced to the problem of existence of\u0000certain sets in the union of these lines.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"122 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209891","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 bounds for ratios of contiguous hypergeometric functions 论连续超几何函数比率的界限
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-08-10 DOI: arxiv-2408.05573
Javier Segura
{"title":"On bounds for ratios of contiguous hypergeometric functions","authors":"Javier Segura","doi":"arxiv-2408.05573","DOIUrl":"https://doi.org/arxiv-2408.05573","url":null,"abstract":"We review recent results on analytical properties (monotonicity and bounds)\u0000for ratios of contiguous functions of hypergeometric type. The cases of\u0000parabolic cylinder functions and modified Bessel functions have been discussed\u0000with considerable detail in the literature, and we give a brief account of\u0000these results, completing some aspects in the case of parabolic cylinder\u0000functions. Different techniques for obtaining these bounds are considered. They\u0000are all based on simple qualitative descriptions of the solutions of associated\u0000ODEs (mainly Riccati equations, but not only Riccati). In spite of their\u0000simplicity, they provide the most accurate global bounds known so far. We also\u0000provide examples of application of these ideas to the more general cases of the\u0000Kummer confluent function and the Gauss hypergeometric function. The function\u0000ratios described in this paper are important functions appearing in a large\u0000number of applications, in which simple approximations are very often required.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209890","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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