The biorthogonal rational functions of the ${_3}F_2$ type on the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials. These properties are described by three difference operators $X,Y,Z$ which are tridiagonal with respect to three distinct bases of the relevant finite-dimensional space. The pairwise commutators of the operators $X,Y,Z$ generate a quadratic algebra which is akin to the algebras of Askey-Wilson type attached to hypergeometric polynomials.
{"title":"An algebraic description of the bispectrality of the biorthogonal rational functions of Hahn type","authors":"S. Tsujimoto, L. Vinet, A. Zhedanov","doi":"10.1090/proc/15225","DOIUrl":"https://doi.org/10.1090/proc/15225","url":null,"abstract":"The biorthogonal rational functions of the ${_3}F_2$ type on the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials. These properties are described by three difference operators $X,Y,Z$ which are tridiagonal with respect to three distinct bases of the relevant finite-dimensional space. The pairwise commutators of the operators $X,Y,Z$ generate a quadratic algebra which is akin to the algebras of Askey-Wilson type attached to hypergeometric polynomials.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83750094","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 : 2020-05-04DOI: 10.31392/MFAT-NPU26_2.2020.04
H. Masliuk, O. Pelekhata, V. Soldatov
We consider a wide class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the normed complex space $(C^{(n)})^m$ of $ngeq r$ times continuously differentiable functions $y:[a,b]tomathbb{C}^{m}$. The boundary conditions for these problems are of the most general form $By=q$, where $B$ is an arbitrary continuous linear operator from $(C^{(n)})^{m}$ to $mathbb{C}^{rm}$. We prove that the solutions to the considered problems can be approximated in $(C^{(n)})^m$ by solutions to some multipoint boundary-value problems. The latter problems do not depend on the right-hand sides of the considered problem and are built explicitly.
{"title":"Approximation properties of multipoint boundary-value problems","authors":"H. Masliuk, O. Pelekhata, V. Soldatov","doi":"10.31392/MFAT-NPU26_2.2020.04","DOIUrl":"https://doi.org/10.31392/MFAT-NPU26_2.2020.04","url":null,"abstract":"We consider a wide class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the normed complex space $(C^{(n)})^m$ of $ngeq r$ times continuously differentiable functions $y:[a,b]tomathbb{C}^{m}$. The boundary conditions for these problems are of the most general form $By=q$, where $B$ is an arbitrary continuous linear operator from $(C^{(n)})^{m}$ to $mathbb{C}^{rm}$. We prove that the solutions to the considered problems can be approximated in $(C^{(n)})^m$ by solutions to some multipoint boundary-value problems. The latter problems do not depend on the right-hand sides of the considered problem and are built explicitly.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79731870","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 a Hilfer fractional differential equation with nonlocal Erdelyi-Kober fractional integral boundary conditions. The existence, uniqueness and Ulam-Hyers stability results are investigated by means of the Krasnoselskii's fixed point theorem and Banach's fixed point theorem. An example is given to illustrate the main results.
{"title":"On a Hilfer fractional differential equation with nonlocal Erdélyi-Kober fractional integral boundary conditions","authors":"M. Abbas","doi":"10.2298/FIL2009003A","DOIUrl":"https://doi.org/10.2298/FIL2009003A","url":null,"abstract":"We consider a Hilfer fractional differential equation with nonlocal Erdelyi-Kober fractional integral boundary conditions. The existence, uniqueness and Ulam-Hyers stability results are investigated by means of the Krasnoselskii's fixed point theorem and Banach's fixed point theorem. An example is given to illustrate the main results.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89389974","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 construct a fundamental system of a $q$-difference Lax pair of rank $N$ in terms of 5d Nekrasov functions with $q=t$. Our fundamental system degenerates by the limit $qto 1$ to a fundamental system of a differential Lax pair, which yields the Fuji-Suzuki-Tsuda system. We introduce tau functions of our system as Fourier transforms of 5d Nekrasov functions. Using asymptotic expansions of the fundamental system at $0$ and $infty$, we obtain several determinantal identities of the tau functions.
{"title":"On $q$-isomonodromic deformations and $q$-Nekrasov functions","authors":"H. Nagoya","doi":"10.3842/SIGMA.2021.050","DOIUrl":"https://doi.org/10.3842/SIGMA.2021.050","url":null,"abstract":"We construct a fundamental system of a $q$-difference Lax pair of rank $N$ in terms of 5d Nekrasov functions with $q=t$. Our fundamental system degenerates by the limit $qto 1$ to a fundamental system of a differential Lax pair, which yields the Fuji-Suzuki-Tsuda system. We introduce tau functions of our system as Fourier transforms of 5d Nekrasov functions. Using asymptotic expansions of the fundamental system at $0$ and $infty$, we obtain several determinantal identities of the tau functions.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84469614","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}
In this paper an asymptotic formula is given for the Lebesgue constants generated by the anisotropically dilated $d$-dimensional simplex. Contrary to many preceding results established only in dimension two, the obtained ones are proved in any dimension. Also, the "rational" and "irrational" parts are both united and separated in one formula.
{"title":"Asymptotics of the Lebesgue constants for a $d$-dimensional simplex","authors":"Yurii Kolomoitsev, E. Liflyand","doi":"10.1090/proc/15438","DOIUrl":"https://doi.org/10.1090/proc/15438","url":null,"abstract":"In this paper an asymptotic formula is given for the Lebesgue constants generated by the anisotropically dilated $d$-dimensional simplex. Contrary to many preceding results established only in dimension two, the obtained ones are proved in any dimension. Also, the \"rational\" and \"irrational\" parts are both united and separated in one formula.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84405552","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 : 2020-04-23DOI: 10.1142/S2010326322500137
Zhaoyu Wang, E. Fan
In this paper, we study the strong asymptotic for the orthogonal polynomials and universality associated with singularly perturbed Pollaczek-Jacobi type weight $$w_{p_J2}(x,t)=e^{-frac{t}{x(1-x)}}x^alpha(1-x)^beta, $$ where $t ge 0$, $alpha >0$, $beta >0$ and $x in [0,1].$ Our main results obtained here include two aspects: { I. Strong asymptotics:} We obtain the strong asymptotic expansions for the monic Pollaczek-Jacobi type orthogonal polynomials in different interval $(0,1)$ and outside of interval $mathbb{C}backslash (0,1)$, respectively; Due to the effect of $frac{t}{x(1-x)}$ for varying $t$, different asymptotic behaviors at the hard edge $0$ and $1$ were found with different scaling schemes. Specifically, the uniform asymptotic behavior can be expressed as a Airy function in the neighborhood of point $1$ as $zeta= 2n^2t to infty, nto infty$, while it is given by a Bessel function as $zeta to 0, n to infty$. { II. Universality:} We respectively calculate the limit of the eigenvalue correlation kernel in the bulk of the spectrum and at the both side of hard edge, which will involve a $psi$-functions associated with a particular Painlev$acute{e}$ uppercaseexpandafter{romannumeral3} equation near $x=pm 1$. Further, we also prove the $psi$-funcation can be approximated by a Bessel kernel as $zeta to 0$ compared with a Airy kernel as $zeta to infty$. Our analysis is based on the Deift-Zhou nonlinear steepest descent method for the Riemann-Hilbert problems.
本文研究了奇异摄动Pollaczek-Jacobi型权值$$w_{p_J2}(x,t)=e^{-frac{t}{x(1-x)}}x^alpha(1-x)^beta, $$的正交多项式的强渐近性和普适性，其中$t ge 0$, $alpha >0$, $beta >0$和$x in [0,1].$，得到的主要结果包括两个方面:{1 .强渐近性:}分别在不同区间$(0,1)$和区间$mathbb{C}backslash (0,1)$外得到了一元Pollaczek-Jacobi型正交多项式的强渐近展开式;由于$frac{t}{x(1-x)}$对$t$的影响，不同标度方案在硬边$0$和$1$处的渐近行为不同。具体地说，一致渐近行为可以表示为在$1$点附近的Airy函数为$zeta= 2n^2t to infty, nto infty$，而由Bessel函数为$zeta to 0, n to infty$给出。{2通用性:}我们分别计算了特征值相关核在光谱主体和硬边两侧的极限，这将涉及与$x=pm 1$附近的特定Painlev $acute{e}$uppercaseexpandafter{romannumeral3}方程相关的$psi$ -函数。此外，我们还证明了$psi$ -函数可以用贝塞尔核近似为$zeta to 0$，而用艾里核近似为$zeta to infty$。我们的分析是基于Deift-Zhou非线性最陡下降法的黎曼-希尔伯特问题。
{"title":"Critical edge behavior in the singularly perturbed Pollaczek–Jacobi type unitary ensemble","authors":"Zhaoyu Wang, E. Fan","doi":"10.1142/S2010326322500137","DOIUrl":"https://doi.org/10.1142/S2010326322500137","url":null,"abstract":"In this paper, we study the strong asymptotic for the orthogonal polynomials and universality associated with singularly perturbed Pollaczek-Jacobi type weight $$w_{p_J2}(x,t)=e^{-frac{t}{x(1-x)}}x^alpha(1-x)^beta, $$ where $t ge 0$, $alpha >0$, $beta >0$ and $x in [0,1].$ Our main results obtained here include two aspects: \u0000{ I. Strong asymptotics:} We obtain the strong asymptotic expansions for the monic Pollaczek-Jacobi type orthogonal polynomials in different interval $(0,1)$ and outside of interval $mathbb{C}backslash (0,1)$, respectively; Due to the effect of $frac{t}{x(1-x)}$ for varying $t$, different asymptotic behaviors at the hard edge $0$ and $1$ were found with different scaling schemes. Specifically, the uniform asymptotic behavior can be expressed as a Airy function in the neighborhood of point $1$ as $zeta= 2n^2t to infty, nto infty$, while it is given by a Bessel function as $zeta to 0, n to infty$. \u0000{ II. Universality:} We respectively calculate the limit of the eigenvalue correlation kernel in the bulk of the spectrum and at the both side of hard edge, which will involve a $psi$-functions associated with a particular Painlev$acute{e}$ uppercaseexpandafter{romannumeral3} equation near $x=pm 1$. Further, we also prove the $psi$-funcation can be approximated by a Bessel kernel as $zeta to 0$ compared with a Airy kernel as $zeta to infty$. Our analysis is based on the Deift-Zhou nonlinear steepest descent method for the Riemann-Hilbert problems.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87629633","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}
It is well-known that Littlewood-Paley operators formed with respect to lacunary sets of finite order are bounded on $L^p (mathbb{R})$ for all $1 l } $.
{"title":"Sharp asymptotic estimates for a class of Littlewood–Paley operators","authors":"Odysseas N. Bakas","doi":"10.4064/SM200514-6-10","DOIUrl":"https://doi.org/10.4064/SM200514-6-10","url":null,"abstract":"It is well-known that Littlewood-Paley operators formed with respect to lacunary sets of finite order are bounded on $L^p (mathbb{R})$ for all $1 l } $.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"180 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83003752","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}
L. Brandolini, L. Colzani, S. Robins, G. Travaglini
We prove an Euler-Maclaurin formula for double polygonal sums and, as a corollary, we obtain approximate quadrature formulas for integrals of smooth functions over polygons with integer vertices. Our Euler-Maclaurin formula is in the spirit of Pick's theorem on the number of integer points in an integer polygon and involves weighted Riemann sums, using tools from Harmonic analysis. Finally, we also exhibit a classical trick, dating back to Huygens and Newton, to accelerate convergence of these Riemann sums.
{"title":"An Euler-MacLaurin formula for polygonal sums","authors":"L. Brandolini, L. Colzani, S. Robins, G. Travaglini","doi":"10.1090/TRAN/8462","DOIUrl":"https://doi.org/10.1090/TRAN/8462","url":null,"abstract":"We prove an Euler-Maclaurin formula for double polygonal sums and, as a corollary, we obtain approximate quadrature formulas for integrals of smooth functions over polygons with integer vertices. Our Euler-Maclaurin formula is in the spirit of Pick's theorem on the number of integer points in an integer polygon and involves weighted Riemann sums, using tools from Harmonic analysis. Finally, we also exhibit a classical trick, dating back to Huygens and Newton, to accelerate convergence of these Riemann sums.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"2017 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89911727","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}
Alexander Barron, M. Erdogan, Terence L. J. Harris
Let $mu$ be an $alpha$-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform $widehat{mu}$. More precisely, if $mathbb{H}$ is a truncated hyperbolic paraboloid in $mathbb{R}^d$ we study the optimal $beta$ for which $$int_{mathbb{H}} |hat{mu}(Rxi)|^2 , d sigma (xi)leq C(alpha, mu) R^{-beta}$$ for all $R > 1$. Our estimates for $beta$ depend on the minimum between the number of positive and negative principal curvatures of $mathbb{H}$; if this number is as large as possible our estimates are sharp in all dimensions.
{"title":"Fourier decay of fractal measures on hyperboloids","authors":"Alexander Barron, M. Erdogan, Terence L. J. Harris","doi":"10.1090/tran/8283","DOIUrl":"https://doi.org/10.1090/tran/8283","url":null,"abstract":"Let $mu$ be an $alpha$-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform $widehat{mu}$. More precisely, if $mathbb{H}$ is a truncated hyperbolic paraboloid in $mathbb{R}^d$ we study the optimal $beta$ for which $$int_{mathbb{H}} |hat{mu}(Rxi)|^2 , d sigma (xi)leq C(alpha, mu) R^{-beta}$$ for all $R > 1$. Our estimates for $beta$ depend on the minimum between the number of positive and negative principal curvatures of $mathbb{H}$; if this number is as large as possible our estimates are sharp in all dimensions.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88339351","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}
In proving the local $T_b$ Theorem for two weights in one dimension [SaShUT] Sawyer, Shen and Uriarte-Tuero used a basic theorem of Hytonen [Hy] to deal with estimates for measures living in adjacent intervals. Hytonen's theorem states that the off-testing condition for the Hilbert transform is controlled by the Muckenhoupt's $A_2$ and $A^*_2$ conditions. So in attempting to extend the two weight $T_b$ theorem to higher dimensions, it is natural to ask if a higher dimensional analogue of Hytonen's theorem holds that permits analogous control of terms involving measures that live on adjacent cubes. In this paper we show that it is not the case even in the presence of the energy conditions used in one dimension [SaShUT]. Thus, in order to obtain a local $T_b$ theorem in higher dimensions, it will be necessary to find some substantially new arguments to control the notoriously difficult nearby form. More precisely, we show that Hytonen's off-testing condition for the two weight fractional integral and the Riesz transform inequalities is not controlled by Muckenhoupt's $A_2^alpha$ and $A_2^{alpha,*}$ conditions and energy conditions.
{"title":"Counterexample to the off-testing condition\u0000in two dimensions","authors":"C. Grigoriadis, M. Paparizos","doi":"10.4064/CM8405-1-2021","DOIUrl":"https://doi.org/10.4064/CM8405-1-2021","url":null,"abstract":"In proving the local $T_b$ Theorem for two weights in one dimension [SaShUT] Sawyer, Shen and Uriarte-Tuero used a basic theorem of Hytonen [Hy] to deal with estimates for measures living in adjacent intervals. Hytonen's theorem states that the off-testing condition for the Hilbert transform is controlled by the Muckenhoupt's $A_2$ and $A^*_2$ conditions. So in attempting to extend the two weight $T_b$ theorem to higher dimensions, it is natural to ask if a higher dimensional analogue of Hytonen's theorem holds that permits analogous control of terms involving measures that live on adjacent cubes. In this paper we show that it is not the case even in the presence of the energy conditions used in one dimension [SaShUT]. Thus, in order to obtain a local $T_b$ theorem in higher dimensions, it will be necessary to find some substantially new arguments to control the notoriously difficult nearby form. More precisely, we show that Hytonen's off-testing condition for the two weight fractional integral and the Riesz transform inequalities is not controlled by Muckenhoupt's $A_2^alpha$ and $A_2^{alpha,*}$ conditions and energy conditions.","PeriodicalId":8451,"journal":{"name":"arXiv: Classical Analysis and ODEs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77098329","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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