In this paper we consider the approximation for functions in some subspaces of L^2 by spherical means of their Fourier integrals and Fourier series on set of full measure. Two main theorems are obtained.
New higher-dimensional distributions have been introduced in the framework of Clifford analysis in previous papers by Brackx, Delanghe and Sommen. Those distributions were defined using spherical co-ordinates, the ＂f...New higher-dimensional distributions have been introduced in the framework of Clifford analysis in previous papers by Brackx, Delanghe and Sommen. Those distributions were defined using spherical co-ordinates, the ＂finite part＂ distribution Fp x＋^μ on the real line and the generalized spherical means involving vector-valued spherical monogenics. In this paper, we make a second generalization, leading to new families of distributions, based on the generalized spherical means involving a multivector-valued spherical monogenic. At the same time, as a result of our attempt at keeping the paper self-contained, it offers an overview of the results found so far.展开更多
We have pointed out in [1] that so far the L^2 norm inequalities with power weights for the Riesz means σ_R~δ(g)(x) of multiple Fourier integrals have been obtained only by Hirschman and J. L. Rubio de Francia respe...We have pointed out in [1] that so far the L^2 norm inequalities with power weights for the Riesz means σ_R~δ(g)(x) of multiple Fourier integrals have been obtained only by Hirschman and J. L. Rubio de Francia respectively:展开更多
Using reproducing kernels for Hilbert spaces, we give best approximation for Weierstrass transform associated with spherical mean operator. Also, estimates of extremal functions are checked.
In this article,we study the boundedness properties of the averaging operator S_(t)^(γ) on Triebel-Lizorkin spaces F_(p,q)^(α)(R^(n))for various p,q.As an application,we obtain the norm convergence rate for S_(t)^(...In this article,we study the boundedness properties of the averaging operator S_(t)^(γ) on Triebel-Lizorkin spaces F_(p,q)^(α)(R^(n))for various p,q.As an application,we obtain the norm convergence rate for S_(t)^(γ)(f)on Triebel-Lizorkin spaces and the relation between the smoothness imposed on functions and the rate of norm convergence of S_(t)^(γ) is given.展开更多
The authors consider the Cauchy problem for the following nonlinear wave equationswhere x ∈ R3, t ≥ 0, ε > 0 is a small parameter, and obtain the sharp bounds for the lifespan of solution to (0.1). Specially, it...The authors consider the Cauchy problem for the following nonlinear wave equationswhere x ∈ R3, t ≥ 0, ε > 0 is a small parameter, and obtain the sharp bounds for the lifespan of solution to (0.1). Specially, it is proved that there exist two constants C1 and C2, which are independent of ε, then the lifespan T(ε) satisfies the folowing inequalities展开更多
In the paper,we study derivative estimates of the iterated spherical averages(At)^(N)(f).We obtain the optimal range of exponents(α,N,p)to ensure the L^(p)boundedness of P(■/■x)(A1)^(N)(f)for 1≤p≤∞,where P is a ...In the paper,we study derivative estimates of the iterated spherical averages(At)^(N)(f).We obtain the optimal range of exponents(α,N,p)to ensure the L^(p)boundedness of P(■/■x)(A1)^(N)(f)for 1≤p≤∞,where P is a homogeneous polynomial of degree a.The main theorem extends some known results.As an application,we obtain the smallest N such that(A1)^(N):L^(p)(R^(n))→L_(α)^(p)(R^(n)).展开更多
A comprehensive review of recent advances in study on thermodynamic models for real electrolyte solutions is presented. The differences between primitive and non-primitive electrolyte models are demonstrated. Some new...A comprehensive review of recent advances in study on thermodynamic models for real electrolyte solutions is presented. The differences between primitive and non-primitive electrolyte models are demonstrated. Some new thermodynamic models for electrolyte solutions based on the mean spherical approximation and perturbation theory are introduced. An extended scaled-particle theory and modified CleggPitzer equation are presented for physical and chemical absorption processes with mixed solvents, respectively. A pseudo one-component two-Yukawa equation of state is used for the aqueous two-phase extraction process in charged colloidal systems.展开更多
Based on statistical mechanics, a review of recent theoretical studies of real electrolyte solutions is presented from three aspects, namely, molecular simulation, mean spherical approximation (MSA), and perturbation ...Based on statistical mechanics, a review of recent theoretical studies of real electrolyte solutions is presented from three aspects, namely, molecular simulation, mean spherical approximation (MSA), and perturbation theory. Recent advances in studies of three kinds of electrostatic potentials of mean force, three kinds of internal energies (ion-ion, ion-dipole, and dipole-dipole interactions), and three kinds of electro-lyte models (primitive, non-primitive, and solvent primitive models) are introduced. The advantages and dis-advantages between primitive and non-primitive models, and between MSA and perturbation theory are dis-cussed. Some new equations of state (EOSs) based on MSA and perturbation theory for real electrolyte so-lutions are introduced. The one-Yukawa EOS and the two-Yukawa EOS for charged colloid systems are presented.展开更多
This paper is an extension of the author's paper (Xie,1982) to the spherical earth.It is found that the simi- lar results are obtained under the assumption of isotropic distribution of horizontal kinetic energy al...This paper is an extension of the author's paper (Xie,1982) to the spherical earth.It is found that the simi- lar results are obtained under the assumption of isotropic distribution of horizontal kinetic energy along the zonal and meridional directions.It points out the limitation of the results already obtained and paves the path for the approach of anisotropie semi-eddy or quasi-eddy motion of the atmosphere.展开更多
文摘In this paper we consider the approximation for functions in some subspaces of L^2 by spherical means of their Fourier integrals and Fourier series on set of full measure. Two main theorems are obtained.
文摘New higher-dimensional distributions have been introduced in the framework of Clifford analysis in previous papers by Brackx, Delanghe and Sommen. Those distributions were defined using spherical co-ordinates, the ＂finite part＂ distribution Fp x＋^μ on the real line and the generalized spherical means involving vector-valued spherical monogenics. In this paper, we make a second generalization, leading to new families of distributions, based on the generalized spherical means involving a multivector-valued spherical monogenic. At the same time, as a result of our attempt at keeping the paper self-contained, it offers an overview of the results found so far.
文摘We have pointed out in [1] that so far the L^2 norm inequalities with power weights for the Riesz means σ_R~δ(g)(x) of multiple Fourier integrals have been obtained only by Hirschman and J. L. Rubio de Francia respectively:
文摘Using reproducing kernels for Hilbert spaces, we give best approximation for Weierstrass transform associated with spherical mean operator. Also, estimates of extremal functions are checked.
基金Supported by the National Natural Science Foundation of China(12071437,12101562)the Natural Science Foundation of Zhejiang(LQ20A010003)the Natural Science Foundation from the Education Department of Anhui Province(KJ2017A847).
文摘In this article,we study the boundedness properties of the averaging operator S_(t)^(γ) on Triebel-Lizorkin spaces F_(p,q)^(α)(R^(n))for various p,q.As an application,we obtain the norm convergence rate for S_(t)^(γ)(f)on Triebel-Lizorkin spaces and the relation between the smoothness imposed on functions and the rate of norm convergence of S_(t)^(γ) is given.
基金This work was supported by South-West Jiaotong University Foundation
文摘The authors consider the Cauchy problem for the following nonlinear wave equationswhere x ∈ R3, t ≥ 0, ε > 0 is a small parameter, and obtain the sharp bounds for the lifespan of solution to (0.1). Specially, it is proved that there exist two constants C1 and C2, which are independent of ε, then the lifespan T(ε) satisfies the folowing inequalities
基金Supported by National Natural Science Foundation of China (Grant Nos. 11771388, 11371316)the Natural Science Foundation of Zhejiang (Grant No. LQ20A010003)
文摘In the paper,we study derivative estimates of the iterated spherical averages(At)^(N)(f).We obtain the optimal range of exponents(α,N,p)to ensure the L^(p)boundedness of P(■/■x)(A1)^(N)(f)for 1≤p≤∞,where P is a homogeneous polynomial of degree a.The main theorem extends some known results.As an application,we obtain the smallest N such that(A1)^(N):L^(p)(R^(n))→L_(α)^(p)(R^(n)).
基金Supported by the Fundamental Research Fund of Tsinghua Universitof China (No. JZ 2002003)
文摘A comprehensive review of recent advances in study on thermodynamic models for real electrolyte solutions is presented. The differences between primitive and non-primitive electrolyte models are demonstrated. Some new thermodynamic models for electrolyte solutions based on the mean spherical approximation and perturbation theory are introduced. An extended scaled-particle theory and modified CleggPitzer equation are presented for physical and chemical absorption processes with mixed solvents, respectively. A pseudo one-component two-Yukawa equation of state is used for the aqueous two-phase extraction process in charged colloidal systems.
基金the Fundamental Research Fund of Tsinghua University of China (No. JZ 2002003)
文摘Based on statistical mechanics, a review of recent theoretical studies of real electrolyte solutions is presented from three aspects, namely, molecular simulation, mean spherical approximation (MSA), and perturbation theory. Recent advances in studies of three kinds of electrostatic potentials of mean force, three kinds of internal energies (ion-ion, ion-dipole, and dipole-dipole interactions), and three kinds of electro-lyte models (primitive, non-primitive, and solvent primitive models) are introduced. The advantages and dis-advantages between primitive and non-primitive models, and between MSA and perturbation theory are dis-cussed. Some new equations of state (EOSs) based on MSA and perturbation theory for real electrolyte so-lutions are introduced. The one-Yukawa EOS and the two-Yukawa EOS for charged colloid systems are presented.
文摘This paper is an extension of the author's paper (Xie,1982) to the spherical earth.It is found that the simi- lar results are obtained under the assumption of isotropic distribution of horizontal kinetic energy along the zonal and meridional directions.It points out the limitation of the results already obtained and paves the path for the approach of anisotropie semi-eddy or quasi-eddy motion of the atmosphere.
