We consider the integral $int_0^inftyleft(frac{sin x}{x}right)^n;dx$ as a function of the positive integer $n$. We show that there exists an asymptotic series in $frac{1}{n}$ and compute the first terms of this series together with an explicit error bound.
{"title":"Asymptotic evaluation of $int_0^inftyleft(frac{sin x}{x}right)^n;dx$","authors":"J. Schlage-Puchta","doi":"10.4134/CKMS.c200133","DOIUrl":"https://doi.org/10.4134/CKMS.c200133","url":null,"abstract":"We consider the integral $int_0^inftyleft(frac{sin x}{x}right)^n;dx$ as a function of the positive integer $n$. We show that there exists an asymptotic series in $frac{1}{n}$ and compute the first terms of this series together with an explicit error bound.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81461878","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 recast Byerly's formula for integrals of products of Legendre polynomials. Then we adopt the idea to the case of Jacobi polynomials. After that, we use the formula to derive an asymptotic formula for integrals of products of Jacobi polynomials. The asymptotic formula is similar to an analogous one recently obtained by the first author and Jeff Geronimo for a different case. Thus, it suggests that such an asymptotic behavior is rather generic for integrals of products of orthogonal polynomials.
{"title":"An Asymptotic Formula for Integrals of Products of Jacobi Polynomials","authors":"Maxim S. Derevyagin, Nicholas Juricic","doi":"10.31390/JOSA.1.4.08","DOIUrl":"https://doi.org/10.31390/JOSA.1.4.08","url":null,"abstract":"We recast Byerly's formula for integrals of products of Legendre polynomials. Then we adopt the idea to the case of Jacobi polynomials. After that, we use the formula to derive an asymptotic formula for integrals of products of Jacobi polynomials. The asymptotic formula is similar to an analogous one recently obtained by the first author and Jeff Geronimo for a different case. Thus, it suggests that such an asymptotic behavior is rather generic for integrals of products of orthogonal polynomials.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81178105","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 give a new proof of Aubin's improvement of the Sobolev inequality on $mathbb{S}^{n}$ under the vanishing of first order moments of the area element and generalize it to higher order moments case. By careful study of an extremal problem on $mathbb{S}^{n}$, we determine the constant explicitly in the second order moments case.
{"title":"Improved Sobolev Inequality Under Constraints","authors":"Fengbo Hang, Xiaodong Wang","doi":"10.1093/IMRN/RNAB067","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB067","url":null,"abstract":"We give a new proof of Aubin's improvement of the Sobolev inequality on $mathbb{S}^{n}$ under the vanishing of first order moments of the area element and generalize it to higher order moments case. By careful study of an extremal problem on $mathbb{S}^{n}$, we determine the constant explicitly in the second order moments case.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91045556","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 aim of this paper is to present a new simple recurrence for Appell and Sheffer sequences in terms of the linear functional that defines them, and to explain how this is equivalent to several well-known characterizations appearing in the literature. We also give several examples, including integral representations of the inverse operators associated to Bernoulli and Euler polynomials, and a new integral representation of the re-scaled Hermite $d$-orthogonal polynomials generalizing the Weierstrass operator related to the Hermite polynomials.
{"title":"Appell and Sheffer sequences: on their characterizations through functionals and examples","authors":"S. A. Carrillo, Miguel Hurtado","doi":"10.5802/CRMATH.172","DOIUrl":"https://doi.org/10.5802/CRMATH.172","url":null,"abstract":"The aim of this paper is to present a new simple recurrence for Appell and Sheffer sequences in terms of the linear functional that defines them, and to explain how this is equivalent to several well-known characterizations appearing in the literature. We also give several examples, including integral representations of the inverse operators associated to Bernoulli and Euler polynomials, and a new integral representation of the re-scaled Hermite $d$-orthogonal polynomials generalizing the Weierstrass operator related to the Hermite polynomials.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74358169","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 presents a somewhat exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric equation (CFGHE) about the fractional regular singular points $x=1$ and $x=infty$. Next, various generating functions of the CFGHF are established. We also develop some differential forms for the CFGHF. Subsequently, differential operators and the contiguous relations are reported. Furthermore, we introduce the conformable fractional integral representation and the fractional Laplace transform of CFGHF. As an application, and after making a suitable change of the independent variable, we provide general solutions of some known conformable fractional differential equations, which could be written by means of the CFGHF.
{"title":"Further study on the conformable fractional Gauss hypergeometric function","authors":"M. Abul-Ez, M. Zayed, Ali Youssef","doi":"10.3934/math.2021588","DOIUrl":"https://doi.org/10.3934/math.2021588","url":null,"abstract":"This paper presents a somewhat exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric equation (CFGHE) about the fractional regular singular points $x=1$ and $x=infty$. Next, various generating functions of the CFGHF are established. We also develop some differential forms for the CFGHF. Subsequently, differential operators and the contiguous relations are reported. Furthermore, we introduce the conformable fractional integral representation and the fractional Laplace transform of CFGHF. As an application, and after making a suitable change of the independent variable, we provide general solutions of some known conformable fractional differential equations, which could be written by means of the CFGHF.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80723284","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 say that a function $f in L^1(mathbb{R})$ tiles at level $w$ by a discrete translation set $Lambda subset mathbb{R}$, if we have $sum_{lambda in Lambda} f(x-lambda)=w$ a.e. In this paper we survey the main results, and prove several new ones, on the structure of tilings of $mathbb{R}$ by translates of a function. The phenomena discussed include tilings of bounded and of unbounded density, uniform distribution of the translates, periodic and non-periodic tilings, and tilings at level zero. Fourier analysis plays an important role in the proofs. Some open problems are also given.
如果我们有$sum_{lambda in Lambda} f(x-lambda)=w$ a.e，我们说一个函数$f in L^1(mathbb{R})$通过一个离散平移集$Lambda subset mathbb{R}$在$w$层上进行平移。在本文中，我们回顾了主要的结果，并证明了几个新的结果，关于通过函数的平移来对$mathbb{R}$层进行平移的结构。讨论的现象包括有界密度和无界密度的平铺、平铺的均匀分布、周期平铺和非周期平铺以及零能级平铺。傅里叶分析在证明中起着重要的作用。给出了一些开放问题。
{"title":"Tiling by translates of a function: results and open problems","authors":"M. N. Kolountzakis, Nir Lev","doi":"10.19086/da.28122","DOIUrl":"https://doi.org/10.19086/da.28122","url":null,"abstract":"We say that a function $f in L^1(mathbb{R})$ tiles at level $w$ by a discrete translation set $Lambda subset mathbb{R}$, if we have $sum_{lambda in Lambda} f(x-lambda)=w$ a.e. In this paper we survey the main results, and prove several new ones, on the structure of tilings of $mathbb{R}$ by translates of a function. The phenomena discussed include tilings of bounded and of unbounded density, uniform distribution of the translates, periodic and non-periodic tilings, and tilings at level zero. Fourier analysis plays an important role in the proofs. Some open problems are also given.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72562600","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 establish a capacitary strong type inequality which resolves a special case of a conjecture by David R. Adams. As a consequence, we obtain several equivalent norms for Choquet integrals associated to Bessel or Riesz capacities. This enables us to obtain bounds for the Hardy-Littlewood maximal function in a sublinear setting.
我们建立了一个容强型不等式，它解决了David R. Adams猜想的一个特例。因此，我们得到了与Bessel或Riesz能力相关的Choquet积分的几个等价范数。这使我们得到了次线性环境下Hardy-Littlewood极大函数的界。
{"title":"On a capacitary strong type inequality and related capacitary estimates","authors":"Keng Hao Ooi, N. Phuc","doi":"10.4171/rmi/1285","DOIUrl":"https://doi.org/10.4171/rmi/1285","url":null,"abstract":"We establish a capacitary strong type inequality which resolves a special case of a conjecture by David R. Adams. As a consequence, we obtain several equivalent norms for Choquet integrals associated to Bessel or Riesz capacities. This enables us to obtain bounds for the Hardy-Littlewood maximal function in a sublinear setting.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90274158","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 an over-determined Falconer type problem on $(k+1)$-point configurations in the plane using the group action framework introduced in cite{GroupAction}. We define the area type of a $(k+1)$-point configuration in the plane to be the vector in $R^{binom{k+1}{2}}$ with entries given by the areas of parallelograms spanned by each pair of points in the configuration. We show that the space of all area types is $2k-1$ dimensional, and prove that a compact set $EsubsetR^d$ of sufficiently large Hausdorff dimension determines a positve measure set of area types.
{"title":"Areas spanned by point configurations in the plane","authors":"Alex McDonald","doi":"10.1090/proc/15348","DOIUrl":"https://doi.org/10.1090/proc/15348","url":null,"abstract":"We consider an over-determined Falconer type problem on $(k+1)$-point configurations in the plane using the group action framework introduced in cite{GroupAction}. We define the area type of a $(k+1)$-point configuration in the plane to be the vector in $R^{binom{k+1}{2}}$ with entries given by the areas of parallelograms spanned by each pair of points in the configuration. We show that the space of all area types is $2k-1$ dimensional, and prove that a compact set $EsubsetR^d$ of sufficiently large Hausdorff dimension determines a positve measure set of area types.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86780963","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}
{"title":"On separated bump conditions for Calderón-Zygmund operators","authors":"A. Lerner","doi":"10.1090/proc/15712","DOIUrl":"https://doi.org/10.1090/proc/15712","url":null,"abstract":"We improve bump conditions for the two-weight boundedness of Calderon-Zygmund operators introduced recently by R. Rahm and S. Spencer.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75437351","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 introduce the Menchov-Rademacher operator for right triangles - a sample two-dimensional maximal operator, and prove an upper bound for its $L_2$ norm.
{"title":"An upper bound for the Menchov-Rademacher operator for right triangles","authors":"A. Vagharshakyan","doi":"10.1090/proc/15950","DOIUrl":"https://doi.org/10.1090/proc/15950","url":null,"abstract":"We introduce the Menchov-Rademacher operator for right triangles - a sample two-dimensional maximal operator, and prove an upper bound for its $L_2$ norm.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80023646","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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