This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation techn...This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation technique introduced by Tadmor-Tang,an optimal pointwise convergence rate is derived for the vanishing viscosity approximations to the initial-boundary value problem for scalar convex conservation laws,whose weak entropy solution is piecewise C 2 -smooth with interaction of elementary waves and the ...展开更多
We offer a new approach to deal with the pointwise convergence of FourierLaplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Rie...We offer a new approach to deal with the pointwise convergence of FourierLaplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter＇s theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson＇s theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson＇s theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres.展开更多
For bounded or some locally bounded functions f measurable on an interval I there is estimated the rate of convergence of the Durrmeyer-type operators Lnf at those points x∈IntI at which the one-sided limits f(x±...For bounded or some locally bounded functions f measurable on an interval I there is estimated the rate of convergence of the Durrmeyer-type operators Lnf at those points x∈IntI at which the one-sided limits f(x± 0) exist. In the main theorems the Chanturiya's modulus of variation is used.展开更多
In this paper, we discuss the pointwise convergence of conjugate convolution op- erators with some applications to wavelets. Some criteria of convergence at （C, 1） continuous points, Lebesgue points and almost every...In this paper, we discuss the pointwise convergence of conjugate convolution op- erators with some applications to wavelets. Some criteria of convergence at （C, 1） continuous points, Lebesgue points and almost everywhere are established.展开更多
The purpose of this paper is to study the pointwise and almost everywhere convergence of the Cesaro means (C,δ) of Fourier-Jacobi expansions, the main term of the Lebesgue constant of the (C ,8) means for - 1<δ≤...The purpose of this paper is to study the pointwise and almost everywhere convergence of the Cesaro means (C,δ) of Fourier-Jacobi expansions, the main term of the Lebesgue constant of the (C ,8) means for - 1<δ≤ α+1/2 is obtained. With the aid of the generalized translation in terms of Jacobi polynomials, pointwise convergence theorems of the (C,δ) means for δ>α+1/2 and equiconvergence theorems for - 1<δ≤α+1/2 are proved. The analogues of the Lebesgue, Salem and Young theorems of the Cesaro means at the critical index δ = α+1/2 are established.展开更多
In this paper, strong summability of Cesaro means (of critical order) of Fourier-Laplace series on unit sphere is discussed. The Pointwise convergence conditions are established. The results of this paper are anal-ogo...In this paper, strong summability of Cesaro means (of critical order) of Fourier-Laplace series on unit sphere is discussed. The Pointwise convergence conditions are established. The results of this paper are anal-ogous to those of single and multiple Fourier series.展开更多
Pointwise convergence and uniform convergence for wavelet frame series is a new topic. With the help of band-limited dual wavelet frames, this topic is first researched.
The convergence of linear fractional transformations is an important topic in mathematics.We study the pointwise convergence of p-adic Mbius maps,and classify the possibilities of limits of pointwise convergent sequen...The convergence of linear fractional transformations is an important topic in mathematics.We study the pointwise convergence of p-adic Mbius maps,and classify the possibilities of limits of pointwise convergent sequences of Mbius maps acting on the projective line P1(C p),where C p is the completion of the algebraic closure of Q p.We show that if the set of pointwise convergence of a sequence of p-adic Mbius maps contains at least three points,the sequence of p-adic Mbius maps either converges to a p-adic Mbius map on the projective line P1(C p),or converges to a constant on the set of pointwise convergence with one unique exceptional point.This result generalizes the result of Piranian and Thron(1957)to the non-archimedean settings.展开更多
For the two-dimensional Walsh system, Gat and Weisz proved the a.e. convergence of Fejer means σnf of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, ...For the two-dimensional Walsh system, Gat and Weisz proved the a.e. convergence of Fejer means σnf of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, β^-1≤n1/n2 ≤β is provided with some fixed parameter ~ 〉 1. In this paper we generalize the result of Gat and Weisz. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets in order to preserve this convergence property.展开更多
Purpose:A new point of view in the study of impact is introduced.Design/methodology/approach:Using fundamental theorems in real analysis we study the convergence of well-known impact measures.Findings:We show that poi...Purpose:A new point of view in the study of impact is introduced.Design/methodology/approach:Using fundamental theorems in real analysis we study the convergence of well-known impact measures.Findings:We show that pointwise convergence is maintained by all well-known impact bundles(such as the h-,g-,and R-bundle)and that theμ-bundle even maintains uniform convergence.Based on these results,a classification of impact bundles is given.Research limitations:As for all impact studies,it is just impossible to study all measures in depth.Practical implications:It is proposed to include convergence properties in the study of impact measures.Originality/value:This article is the first to present a bundle classification based on convergence properties of impact bundles.展开更多
In this paper,the authors study the almost everywhere pointwise convergence problem along a class of restricted curves in R×R given by{(y,t):y∈Γ(x,t)}for each t∈[0,1],where Γ(x,t)={γ(x,t,θ):θ∈Θ}for a giv...In this paper,the authors study the almost everywhere pointwise convergence problem along a class of restricted curves in R×R given by{(y,t):y∈Γ(x,t)}for each t∈[0,1],where Γ(x,t)={γ(x,t,θ):θ∈Θ}for a given compact set Θ in R of the fractiona Schrodinger propagator and Boussinesq operator.They focus on the relationship between the upper Minkowski dimension of Θ and the optimal s for which limy∈Γ(x,t)(y,t)→(x,0) e^(it(√△)^(a))f(y)=f(x),limy∈Γ(x,t)(y,t)→(x,0) B_(t)f(y)=f(x),a.e.,whenever f∈H^(s)(R).展开更多
In the present paper we state some approximation theorems concerning point- wise convergence and its rate for a class of non-convolution type nonlinear integral opera- tors of the form：Tλ（f;x）=B∫AKλ（t,x,f（t）...In the present paper we state some approximation theorems concerning point- wise convergence and its rate for a class of non-convolution type nonlinear integral opera- tors of the form：Tλ（f;x）=B∫AKλ（t,x,f（t））dr,x∈〈a,b〉λλA.In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 off as （x,λ） → （x0, λ0） in LI 〈A,B 〉, where 〈 a,b 〉 and 〈A,B 〉 are is an arbitrary intervals in R, A is a non-empty set of indices with a topology and X0 an accumulation point of A in this topology. The results of the present paper generalize several ones obtained previously in the papers [191-[23]展开更多
In this paper we shall defin a kind of generahzed Szász-Mirakjan operator and discuss its convergence and degree of approximation,extend some results got by J.Grof and Z.Ditzian.
In this paper, the author define a kind of generalized Szasz-Mirakjan operator and discuss its convergence and degree of the approximation,extend some results got by J. Grof[1] and Z. Ditzian[2].
In this paper we study the convergence in norm and pointwise convergence of wavelet expansion in the Orlicz spaces,and prove that,under certain conditions on the wavelet,the wavelet expansion converges in the Orlicz-n...In this paper we study the convergence in norm and pointwise convergence of wavelet expansion in the Orlicz spaces,and prove that,under certain conditions on the wavelet,the wavelet expansion converges in the Orlicz-norm and also converges almost everywhere.展开更多
This paper is devoted to the study of semi-commutative harmonic analysis associated with Hermite semigroups. In the first part, we establish the noncommutative maximal inequalities for Bochner-Riesz means associated w...This paper is devoted to the study of semi-commutative harmonic analysis associated with Hermite semigroups. In the first part, we establish the noncommutative maximal inequalities for Bochner-Riesz means associated with Hermite operators and then obtain the corresponding pointwise convergence theorems. In particular, we develop a noncommutative version of Stein's theorem of Bochner-Riesz means for Hermite operators. In the second part, we investigate two noncommutative multiplier theorems. Our approach in this part relies on a noncommutative analog of the classical Littlewood-Paley-Stein theory associated with Hermite semigroups.展开更多
In this paper,we establish Schrödinger maximal estimates associated with the finite type phaseФ(ξ_(1),ξ_(2)):=ξ_(1)^(m)+ξ_(2)^(m),where m≥4 is an even number.Following[12],we prove an L2 fractal restriction...In this paper,we establish Schrödinger maximal estimates associated with the finite type phaseФ(ξ_(1),ξ_(2)):=ξ_(1)^(m)+ξ_(2)^(m),where m≥4 is an even number.Following[12],we prove an L2 fractal restriction estimate associated with the surface{(ξ_(1),ξ_(2),Ф(ξ_(1),ξ_(2))):(ξ_(1),ξ_(2)∈[0,]^(2)}as the main result,which also gives results on the average Fourier decay of fractal measures associated with these surfaces.The key ingredients of the proof include the rescaling technique from[16],Bourgain-Demeter’sℓ^(2)decoupling inequality,the reduction of dimension arguments from[17]and induction on scales.We notice that our Theorem 1.1 has some similarities with the results in[8].However,their results do not cover ours.Their arguments depend on the positive definiteness of the Hessian matrix of the phase function,while our phase functions are degenerate.展开更多
Due to the difficulty in obtaining the a priori estimate,it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dime...Due to the difficulty in obtaining the a priori estimate,it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions(2D or 3D).We here propose and analyze finite difference methods for solving the coupled GrossPitaevskii equations in two dimensions,which models the two-component Bose-Einstein condensates with an internal atomic Josephson junction.The methods which we considered include two conservative type schemes and two non-conservative type schemes.Discrete conservation laws and solvability of the schemes are analyzed.For the four proposed finite difference methods,we establish the optimal convergence rates for the error at the order of O(h^2+τ~2)in the l~∞-norm(i.e.,the point-wise error estimates)with the time stepτand the mesh size h.Besides the standard techniques of the energy method,the key techniques in the analysis is to use the cut-off function technique,transformation between the time and space direction and the method of order reduction.All the methods and results here are also valid and can be easily extended to the three-dimensional case.Finally,numerical results are reported to confirm our theoretical error estimates for the numerical methods.展开更多
In this paper, we discuss tightness and fan tightness of multifunction spaces with pointwise convergence topology or compact-open topology, and generalize some results on con- tinuous single-valued function spaces to ...In this paper, we discuss tightness and fan tightness of multifunction spaces with pointwise convergence topology or compact-open topology, and generalize some results on con- tinuous single-valued function spaces to continuous multifunction spaces.展开更多
基金supported by the NSF China#10571075NSF-Guangdong China#04010473+1 种基金The research of the second author was supported by Jinan University Foundation#51204033the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State education Ministry#2005-383
文摘This article is concerned with the pointwise error estimates for vanishing vis- cosity approximations to scalar convex conservation laws with boundary.By the weighted error function and a bootstrap extrapolation technique introduced by Tadmor-Tang,an optimal pointwise convergence rate is derived for the vanishing viscosity approximations to the initial-boundary value problem for scalar convex conservation laws,whose weak entropy solution is piecewise C 2 -smooth with interaction of elementary waves and the ...
基金Sponsored by Research Grant of the University of Macao No. RG024/03-04S/QT/FST
文摘We offer a new approach to deal with the pointwise convergence of FourierLaplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter＇s theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson＇s theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson＇s theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres.
文摘For bounded or some locally bounded functions f measurable on an interval I there is estimated the rate of convergence of the Durrmeyer-type operators Lnf at those points x∈IntI at which the one-sided limits f(x± 0) exist. In the main theorems the Chanturiya's modulus of variation is used.
文摘In this paper, we discuss the pointwise convergence of conjugate convolution op- erators with some applications to wavelets. Some criteria of convergence at （C, 1） continuous points, Lebesgue points and almost everywhere are established.
文摘The purpose of this paper is to study the pointwise and almost everywhere convergence of the Cesaro means (C,δ) of Fourier-Jacobi expansions, the main term of the Lebesgue constant of the (C ,8) means for - 1<δ≤ α+1/2 is obtained. With the aid of the generalized translation in terms of Jacobi polynomials, pointwise convergence theorems of the (C,δ) means for δ>α+1/2 and equiconvergence theorems for - 1<δ≤α+1/2 are proved. The analogues of the Lebesgue, Salem and Young theorems of the Cesaro means at the critical index δ = α+1/2 are established.
文摘In this paper, strong summability of Cesaro means (of critical order) of Fourier-Laplace series on unit sphere is discussed. The Pointwise convergence conditions are established. The results of this paper are anal-ogous to those of single and multiple Fourier series.
文摘Pointwise convergence and uniform convergence for wavelet frame series is a new topic. With the help of band-limited dual wavelet frames, this topic is first researched.
基金supported by National Natural Science Foundation of China(Grant Nos.10831004 and 11301510)
文摘The convergence of linear fractional transformations is an important topic in mathematics.We study the pointwise convergence of p-adic Mbius maps,and classify the possibilities of limits of pointwise convergent sequences of Mbius maps acting on the projective line P1(C p),where C p is the completion of the algebraic closure of Q p.We show that if the set of pointwise convergence of a sequence of p-adic Mbius maps contains at least three points,the sequence of p-adic Mbius maps either converges to a p-adic Mbius map on the projective line P1(C p),or converges to a constant on the set of pointwise convergence with one unique exceptional point.This result generalizes the result of Piranian and Thron(1957)to the non-archimedean settings.
基金Supported by the Scientific Board of College of Nyiregyhaza
文摘For the two-dimensional Walsh system, Gat and Weisz proved the a.e. convergence of Fejer means σnf of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, β^-1≤n1/n2 ≤β is provided with some fixed parameter ~ 〉 1. In this paper we generalize the result of Gat and Weisz. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets in order to preserve this convergence property.
基金Supported by National Natural Science Foundation of China(Grant Nos.10671062,11071065 and 11171306)Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20094306110004)Hu’nan Provincial Natural Science Foundation of China(Grant No.06JJ5012)
文摘In this paper, a new result on pointwise convergence of wavelets of generalized Shannon type is proved, which improves a theorem established by Zayed.
基金The author thanks Li Li(National Science Library,CAS)for drawing Figure 1.
文摘Purpose:A new point of view in the study of impact is introduced.Design/methodology/approach:Using fundamental theorems in real analysis we study the convergence of well-known impact measures.Findings:We show that pointwise convergence is maintained by all well-known impact bundles(such as the h-,g-,and R-bundle)and that theμ-bundle even maintains uniform convergence.Based on these results,a classification of impact bundles is given.Research limitations:As for all impact studies,it is just impossible to study all measures in depth.Practical implications:It is proposed to include convergence properties in the study of impact measures.Originality/value:This article is the first to present a bundle classification based on convergence properties of impact bundles.
基金supported by the National Natural Science Foundation of China(No.12071052)the Research Foundation for Youth Scholars of Beijing Technology and Business University(No.QNJJ2021-02)。
文摘In this paper,the authors study the almost everywhere pointwise convergence problem along a class of restricted curves in R×R given by{(y,t):y∈Γ(x,t)}for each t∈[0,1],where Γ(x,t)={γ(x,t,θ):θ∈Θ}for a given compact set Θ in R of the fractiona Schrodinger propagator and Boussinesq operator.They focus on the relationship between the upper Minkowski dimension of Θ and the optimal s for which limy∈Γ(x,t)(y,t)→(x,0) e^(it(√△)^(a))f(y)=f(x),limy∈Γ(x,t)(y,t)→(x,0) B_(t)f(y)=f(x),a.e.,whenever f∈H^(s)(R).
文摘In the present paper we state some approximation theorems concerning point- wise convergence and its rate for a class of non-convolution type nonlinear integral opera- tors of the form：Tλ（f;x）=B∫AKλ（t,x,f（t））dr,x∈〈a,b〉λλA.In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 off as （x,λ） → （x0, λ0） in LI 〈A,B 〉, where 〈 a,b 〉 and 〈A,B 〉 are is an arbitrary intervals in R, A is a non-empty set of indices with a topology and X0 an accumulation point of A in this topology. The results of the present paper generalize several ones obtained previously in the papers [191-[23]
文摘In this paper we shall defin a kind of generahzed Szász-Mirakjan operator and discuss its convergence and degree of approximation,extend some results got by J.Grof and Z.Ditzian.
文摘In this paper, the author define a kind of generalized Szasz-Mirakjan operator and discuss its convergence and degree of the approximation,extend some results got by J. Grof[1] and Z. Ditzian[2].
基金Supported by the Grant from Chongqing Education Commission (Grant No. KJ061201)
文摘In this paper we study the convergence in norm and pointwise convergence of wavelet expansion in the Orlicz spaces,and prove that,under certain conditions on the wavelet,the wavelet expansion converges in the Orlicz-norm and also converges almost everywhere.
基金supported by National Natural Science Foundation of China (Grant No. 12071355)National Research Foundation of Korea (Grant No. NRF-2022R1A2C1092320)Samsung Science and Technology Foundation (Grant No. SSTF-BA2002-01)。
文摘This paper is devoted to the study of semi-commutative harmonic analysis associated with Hermite semigroups. In the first part, we establish the noncommutative maximal inequalities for Bochner-Riesz means associated with Hermite operators and then obtain the corresponding pointwise convergence theorems. In particular, we develop a noncommutative version of Stein's theorem of Bochner-Riesz means for Hermite operators. In the second part, we investigate two noncommutative multiplier theorems. Our approach in this part relies on a noncommutative analog of the classical Littlewood-Paley-Stein theory associated with Hermite semigroups.
基金Supported by National Natural Science Foundation of China(Grant Nos.12101562,12101040,12271051 and 12371239)by a grant from the China Scholarship Council(CSC)。
文摘In this paper,we establish Schrödinger maximal estimates associated with the finite type phaseФ(ξ_(1),ξ_(2)):=ξ_(1)^(m)+ξ_(2)^(m),where m≥4 is an even number.Following[12],we prove an L2 fractal restriction estimate associated with the surface{(ξ_(1),ξ_(2),Ф(ξ_(1),ξ_(2))):(ξ_(1),ξ_(2)∈[0,]^(2)}as the main result,which also gives results on the average Fourier decay of fractal measures associated with these surfaces.The key ingredients of the proof include the rescaling technique from[16],Bourgain-Demeter’sℓ^(2)decoupling inequality,the reduction of dimension arguments from[17]and induction on scales.We notice that our Theorem 1.1 has some similarities with the results in[8].However,their results do not cover ours.Their arguments depend on the positive definiteness of the Hessian matrix of the phase function,while our phase functions are degenerate.
基金supported by National Natural Science Foundation of China(Grant No.11201239)the Singapore A*STAR SERC PSF(Grant No.1321202067)
文摘Due to the difficulty in obtaining the a priori estimate,it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions(2D or 3D).We here propose and analyze finite difference methods for solving the coupled GrossPitaevskii equations in two dimensions,which models the two-component Bose-Einstein condensates with an internal atomic Josephson junction.The methods which we considered include two conservative type schemes and two non-conservative type schemes.Discrete conservation laws and solvability of the schemes are analyzed.For the four proposed finite difference methods,we establish the optimal convergence rates for the error at the order of O(h^2+τ~2)in the l~∞-norm(i.e.,the point-wise error estimates)with the time stepτand the mesh size h.Besides the standard techniques of the energy method,the key techniques in the analysis is to use the cut-off function technique,transformation between the time and space direction and the method of order reduction.All the methods and results here are also valid and can be easily extended to the three-dimensional case.Finally,numerical results are reported to confirm our theoretical error estimates for the numerical methods.
基金the Science and Research Foundation of Hangzhou Normal University (No.02010180)
文摘In this paper, we discuss tightness and fan tightness of multifunction spaces with pointwise convergence topology or compact-open topology, and generalize some results on con- tinuous single-valued function spaces to continuous multifunction spaces.
