Pub Date : 2021-07-12DOI: 10.1142/s0219530521500305
Al'i Guzm'an Ad'an, I. Sabadini, F. Sommen
In this paper, we obtain a plane wave decomposition for the delta distribution in superspace, provided that the superdimension is not odd and negative. This decomposition allows for explicit inversion formulas for the super Radon transform in these cases. Moreover, we prove a more general Radon inversion formula valid for all possible integer values of the superdimension. The proof of this result comes along with the study of fractional powers of the super Laplacian, their fundamental solutions, and the plane wave decompositions of super Riesz kernels.
{"title":"On the Radon transform and the Dirac delta distribution in superspace","authors":"Al'i Guzm'an Ad'an, I. Sabadini, F. Sommen","doi":"10.1142/s0219530521500305","DOIUrl":"https://doi.org/10.1142/s0219530521500305","url":null,"abstract":"In this paper, we obtain a plane wave decomposition for the delta distribution in superspace, provided that the superdimension is not odd and negative. This decomposition allows for explicit inversion formulas for the super Radon transform in these cases. Moreover, we prove a more general Radon inversion formula valid for all possible integer values of the superdimension. The proof of this result comes along with the study of fractional powers of the super Laplacian, their fundamental solutions, and the plane wave decompositions of super Riesz kernels.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45188166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-06-29DOI: 10.1142/S0219530521500135
Long Huang, F. Weisz, Dachun Yang, Wen Yuan
Let [Formula: see text], [Formula: see text] be the mixed-norm Lebesgue space, and [Formula: see text] an integrable function. In this paper, via establishing the boundedness of the mixed centered Hardy–Littlewood maximal operator [Formula: see text] from [Formula: see text] to itself or to the weak mixed-norm Lebesgue space [Formula: see text] under some sharp assumptions on [Formula: see text] and [Formula: see text], the authors show that the [Formula: see text]-mean of [Formula: see text] converges to [Formula: see text] almost everywhere over the diagonal if the Fourier transform [Formula: see text] of [Formula: see text] belongs to some mixed-norm homogeneous Herz space [Formula: see text] with [Formula: see text] being the conjugate index of [Formula: see text]. Furthermore, by introducing another mixed-norm homogeneous Herz space and establishing a characterization of this Herz space, the authors then extend the above almost everywhere convergence of [Formula: see text]-means to the unrestricted case. Finally, the authors show that the [Formula: see text]-mean of [Formula: see text] converges over the diagonal to [Formula: see text] at all its [Formula: see text]-Lebesgue points if and only if [Formula: see text] belongs to [Formula: see text], and a similar conclusion also holds true for the unrestricted convergence at strong [Formula: see text]-Lebesgue points. Observe that, in all these results, those Herz spaces to which [Formula: see text] belongs prove to be the best choice in some sense.
{"title":"Summability of Fourier transforms on mixed-norm Lebesgue spaces via associated Herz spaces","authors":"Long Huang, F. Weisz, Dachun Yang, Wen Yuan","doi":"10.1142/S0219530521500135","DOIUrl":"https://doi.org/10.1142/S0219530521500135","url":null,"abstract":"Let [Formula: see text], [Formula: see text] be the mixed-norm Lebesgue space, and [Formula: see text] an integrable function. In this paper, via establishing the boundedness of the mixed centered Hardy–Littlewood maximal operator [Formula: see text] from [Formula: see text] to itself or to the weak mixed-norm Lebesgue space [Formula: see text] under some sharp assumptions on [Formula: see text] and [Formula: see text], the authors show that the [Formula: see text]-mean of [Formula: see text] converges to [Formula: see text] almost everywhere over the diagonal if the Fourier transform [Formula: see text] of [Formula: see text] belongs to some mixed-norm homogeneous Herz space [Formula: see text] with [Formula: see text] being the conjugate index of [Formula: see text]. Furthermore, by introducing another mixed-norm homogeneous Herz space and establishing a characterization of this Herz space, the authors then extend the above almost everywhere convergence of [Formula: see text]-means to the unrestricted case. Finally, the authors show that the [Formula: see text]-mean of [Formula: see text] converges over the diagonal to [Formula: see text] at all its [Formula: see text]-Lebesgue points if and only if [Formula: see text] belongs to [Formula: see text], and a similar conclusion also holds true for the unrestricted convergence at strong [Formula: see text]-Lebesgue points. Observe that, in all these results, those Herz spaces to which [Formula: see text] belongs prove to be the best choice in some sense.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":"1 1","pages":"1-50"},"PeriodicalIF":2.2,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43960497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-31DOI: 10.1142/S021953052150010X
Z. Wu, R. Zhang
The nonlinear chaotic differential/algebraic equation (DAE) has been established to simulate the nonuniform oscillations of the motion of a falling sphere in the non-Newtonian fluid. The DAE is obt...
{"title":"Probabilistic solutions to DAEs learning from physical data","authors":"Z. Wu, R. Zhang","doi":"10.1142/S021953052150010X","DOIUrl":"https://doi.org/10.1142/S021953052150010X","url":null,"abstract":"The nonlinear chaotic differential/algebraic equation (DAE) has been established to simulate the nonuniform oscillations of the motion of a falling sphere in the non-Newtonian fluid. The DAE is obt...","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":"1-19"},"PeriodicalIF":2.2,"publicationDate":"2021-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48695686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-31DOI: 10.1142/S0219530521500123
Yunyou Qian, Dansheng Yu
In this paper, we introduce some neural network interpolation operators activated by smooth ramp functions. By using the smoothness of the ramp functions, we can give some useful estimates of the derivatives of the neural networks, which combining with some techniques in approximation theory enable us to establish the converse estimates of approximation by neural networks. We establish both the direct and the converse results of approximation by the new neural network operators defined by us, and thus give the essential approximation rate. To improve the approximation rate for functions of smoothness, we further introduce linear combinations of the new operators. The new combinations interpolate the objective function and its derivative. We also estimate the uniform convergence rate and simultaneous approximation rate by the new combinations.
{"title":"Neural network interpolation operators activated by smooth ramp functions","authors":"Yunyou Qian, Dansheng Yu","doi":"10.1142/S0219530521500123","DOIUrl":"https://doi.org/10.1142/S0219530521500123","url":null,"abstract":"In this paper, we introduce some neural network interpolation operators activated by smooth ramp functions. By using the smoothness of the ramp functions, we can give some useful estimates of the derivatives of the neural networks, which combining with some techniques in approximation theory enable us to establish the converse estimates of approximation by neural networks. We establish both the direct and the converse results of approximation by the new neural network operators defined by us, and thus give the essential approximation rate. To improve the approximation rate for functions of smoothness, we further introduce linear combinations of the new operators. The new combinations interpolate the objective function and its derivative. We also estimate the uniform convergence rate and simultaneous approximation rate by the new combinations.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48866183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-07DOI: 10.1142/s0219530522500038
V'ictor Almeida, J. Betancor, L. Rodr'iguez-Mesa
By {T t }t>0 we denote the semigroup of operators generated by the Friedrichs extension of the Schrödinger operator with the inverse square potential La = −∆+ a |x|2 defined in C∞ c (R n {0}). In this paper we establish weighted L-inequalities for the maximal, variation, oscillation and jump operators associated with {t∂ t T a t }t>0, where α ≥ 0 and ∂ α t denotes the Weyl fractional derivative. The range of values p that works is different when a ≥ 0 and when − (n−2) 2 4 < a < 0.
我们用{T T} T >0表示由Schrödinger算子的friedrichhs扩展生成的算子半群，该算子具有平方逆势La = -∆+ a |x|2，定义在C∞C (R n {0})中。本文建立了与{t∂t ta t}t>0相关的极大算子、变分算子、振荡算子和跳跃算子的加权l不等式，其中α≥0，∂α t表示Weyl分数阶导数。当a≥0和−(n−2)24 < a < 0时，p的取值范围不同。
{"title":"Variation Operators Associated with the Semigroups Generated by Schrodinger Operators with Inverse Square Potentials","authors":"V'ictor Almeida, J. Betancor, L. Rodr'iguez-Mesa","doi":"10.1142/s0219530522500038","DOIUrl":"https://doi.org/10.1142/s0219530522500038","url":null,"abstract":"By {T t }t>0 we denote the semigroup of operators generated by the Friedrichs extension of the Schrödinger operator with the inverse square potential La = −∆+ a |x|2 defined in C∞ c (R n {0}). In this paper we establish weighted L-inequalities for the maximal, variation, oscillation and jump operators associated with {t∂ t T a t }t>0, where α ≥ 0 and ∂ α t denotes the Weyl fractional derivative. The range of values p that works is different when a ≥ 0 and when − (n−2) 2 4 < a < 0.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43672526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-30DOI: 10.1142/S0219530521500056
X. Zhong
We investigate an initial boundary value problem of two-dimensional nonhomogeneous heat conducting magnetohydrodynamic equations. We prove that there exists a unique global strong solution. Moreove...
{"title":"Global existence and large time behavior of strong solutions to the nonhomogeneous heat conducting magnetohydrodynamic equations with large initial data and vacuum","authors":"X. Zhong","doi":"10.1142/S0219530521500056","DOIUrl":"https://doi.org/10.1142/S0219530521500056","url":null,"abstract":"We investigate an initial boundary value problem of two-dimensional nonhomogeneous heat conducting magnetohydrodynamic equations. We prove that there exists a unique global strong solution. Moreove...","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45465619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-19DOI: 10.1142/S0219530521500081
R. Fetecau, Hansol Park, F. Patacchini
We investigate a model for collective behavior with intrinsic interactions on Riemannian manifolds. We establish the well-posedness of measure-valued solutions (defined via mass transport) on sphere, as well as investigate the mean-field particle approximation. We study the long-time behavior of solutions to the model on sphere, where the primary goal is to establish sufficient conditions for a consensus state to form asymptotically. Well-posedness of solutions and the formation of consensus are also investigated for other manifolds (e.g., a hypercylinder).
{"title":"Well-posedness and asymptotic behavior of an aggregation model with intrinsic interactions on sphere and other manifolds","authors":"R. Fetecau, Hansol Park, F. Patacchini","doi":"10.1142/S0219530521500081","DOIUrl":"https://doi.org/10.1142/S0219530521500081","url":null,"abstract":"We investigate a model for collective behavior with intrinsic interactions on Riemannian manifolds. We establish the well-posedness of measure-valued solutions (defined via mass transport) on sphere, as well as investigate the mean-field particle approximation. We study the long-time behavior of solutions to the model on sphere, where the primary goal is to establish sufficient conditions for a consensus state to form asymptotically. Well-posedness of solutions and the formation of consensus are also investigated for other manifolds (e.g., a hypercylinder).","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":"89 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86084935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-10DOI: 10.1142/S0219530521500068
Ning Bi, J. Tan, Wai-Shing Tang
In this paper, we provide a necessary condition and a sufficient condition such that any [Formula: see text]-sparse vector [Formula: see text] can be recovered from [Formula: see text] via [Formula: see text] local minimization. Moreover, we further verify that the sufficient condition is naturally valid when the restricted isometry constant of the measurement matrix [Formula: see text] satisfies [Formula: see text]. Compared with the existing [Formula: see text] local recoverability condition [Formula: see text], this result shows that [Formula: see text] local recoverability contains more measurement matrices.
{"title":"A new sufficient condition for sparse vector recovery via ℓ1 − ℓ2 local minimization","authors":"Ning Bi, J. Tan, Wai-Shing Tang","doi":"10.1142/S0219530521500068","DOIUrl":"https://doi.org/10.1142/S0219530521500068","url":null,"abstract":"In this paper, we provide a necessary condition and a sufficient condition such that any [Formula: see text]-sparse vector [Formula: see text] can be recovered from [Formula: see text] via [Formula: see text] local minimization. Moreover, we further verify that the sufficient condition is naturally valid when the restricted isometry constant of the measurement matrix [Formula: see text] satisfies [Formula: see text]. Compared with the existing [Formula: see text] local recoverability condition [Formula: see text], this result shows that [Formula: see text] local recoverability contains more measurement matrices.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":"1 1","pages":"1-13"},"PeriodicalIF":2.2,"publicationDate":"2021-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49210359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-04-08DOI: 10.1142/s0219530521500202
D. Dai, Luming Yao
In this paper, we consider the discrete Laguerre polynomials [Formula: see text] orthogonal with respect to the weight function [Formula: see text] supported on the infinite nodes [Formula: see text]. We focus on the “band-saturated region” situation when the parameter [Formula: see text]. As [Formula: see text], uniform expansions for [Formula: see text] are achieved for [Formula: see text] in different regions in the complex plane. Typically, the Airy-function expansions and Gamma-function expansions are derived for [Formula: see text] near the endpoints of the band and the origin, respectively. The asymptotics for the normalizing coefficient [Formula: see text], recurrence coefficients [Formula: see text] and [Formula: see text], are also obtained. Our method is based on the Deift–Zhou steepest descent method for Riemann–Hilbert problems.
{"title":"Uniform asymptotics for the discrete Laguerre polynomials","authors":"D. Dai, Luming Yao","doi":"10.1142/s0219530521500202","DOIUrl":"https://doi.org/10.1142/s0219530521500202","url":null,"abstract":"In this paper, we consider the discrete Laguerre polynomials [Formula: see text] orthogonal with respect to the weight function [Formula: see text] supported on the infinite nodes [Formula: see text]. We focus on the “band-saturated region” situation when the parameter [Formula: see text]. As [Formula: see text], uniform expansions for [Formula: see text] are achieved for [Formula: see text] in different regions in the complex plane. Typically, the Airy-function expansions and Gamma-function expansions are derived for [Formula: see text] near the endpoints of the band and the origin, respectively. The asymptotics for the normalizing coefficient [Formula: see text], recurrence coefficients [Formula: see text] and [Formula: see text], are also obtained. Our method is based on the Deift–Zhou steepest descent method for Riemann–Hilbert problems.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47313128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-17DOI: 10.1142/s0219530521500329
Q. Du, T. Mengesha, Xiaochuan Tian
This work aims to prove a Hardy-type inequality and a trace theorem for a class of function spaces on smooth domains with a nonlocal character. Functions in these spaces are allowed to be as rough as an [Formula: see text]-function inside the domain of definition but as smooth as a [Formula: see text]-function near the boundary. This feature is captured by a norm that is characterized by a nonlocal interaction kernel defined heterogeneously with a special localization feature on the boundary. Thus, the trace theorem we obtain here can be viewed as an improvement and refinement of the classical trace theorem for fractional Sobolev spaces [Formula: see text]. Similarly, the Hardy-type inequalities we establish for functions that vanish on the boundary show that functions in this generalized space have the same decay rate to the boundary as functions in the smaller space [Formula: see text]. The results we prove extend existing results shown in the Hilbert space setting with [Formula: see text]. A Poincaré-type inequality we establish for the function space under consideration together with the new trace theorem allows formulating and proving well-posedness of a nonlinear nonlocal variational problem with conventional local boundary condition.
{"title":"Fractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localization","authors":"Q. Du, T. Mengesha, Xiaochuan Tian","doi":"10.1142/s0219530521500329","DOIUrl":"https://doi.org/10.1142/s0219530521500329","url":null,"abstract":"This work aims to prove a Hardy-type inequality and a trace theorem for a class of function spaces on smooth domains with a nonlocal character. Functions in these spaces are allowed to be as rough as an [Formula: see text]-function inside the domain of definition but as smooth as a [Formula: see text]-function near the boundary. This feature is captured by a norm that is characterized by a nonlocal interaction kernel defined heterogeneously with a special localization feature on the boundary. Thus, the trace theorem we obtain here can be viewed as an improvement and refinement of the classical trace theorem for fractional Sobolev spaces [Formula: see text]. Similarly, the Hardy-type inequalities we establish for functions that vanish on the boundary show that functions in this generalized space have the same decay rate to the boundary as functions in the smaller space [Formula: see text]. The results we prove extend existing results shown in the Hilbert space setting with [Formula: see text]. A Poincaré-type inequality we establish for the function space under consideration together with the new trace theorem allows formulating and proving well-posedness of a nonlinear nonlocal variational problem with conventional local boundary condition.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43108895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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