Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quan...Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quantum mechanics and value problems.In this paper,we first give the definition of a quasi-Cauchy type integral in complex Clifford analysis,and get the Plemelj formula for it.Second,we discuss the H?lder continuity for the Cauchy-type integral operators with values in a complex Clifford algebra.Finally,we prove the existence of solutions for a class of linear boundary value problems and give the integral representation for the solution.展开更多
Under the foundation of Cauchy integral formula on certain distinguished boundary for functions with values in universal Clifford algebra, we define the Cauchy type integral with values in a universal Clifford algebra...Under the foundation of Cauchy integral formula on certain distinguished boundary for functions with values in universal Clifford algebra, we define the Cauchy type integral with values in a universal Clifford algebra, obtain its Cauchy principal value and Plemelj formula on certain distinguished boundary.展开更多
In this article, we establish the Gauss Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic me...In this article, we establish the Gauss Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic method. The Cauchy-Pompeiu formula for Clifford-valued functions under the weak condition will be derived as their simple application. Furthermore, Cauchy formula for monogenic functions under the weak condition is derived directly from the Cauchy-Pompeiu formula.展开更多
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
Complex Hermitian Clifford analysis emerged recently as a refinement of the theory of several complex variables, while at the same time, the theory of bicomplex numbers motivated by the bicomplex version of quantum me...Complex Hermitian Clifford analysis emerged recently as a refinement of the theory of several complex variables, while at the same time, the theory of bicomplex numbers motivated by the bicomplex version of quantum mechanics is also under full development. This stimulates us to combine the Hermitian Clifford analysis with the theory of bicomplex number so as to set up the theory of bicomplex Hermitian Clifford analysis. In parallel with the Euclidean Clifford analysis, the bicomplex Hermitian Clifford analysis is centered around the bicomplex Hermitian Dirac operator D: C^∞(R^4n W4n) →4 C^∞(R^4n, W4n), where W4n is the tensor product of three algebras, i.e., the hyperbolic quaternion B, the bicomplex number B, and the Clifford algebra R0,4n. The operator D is a square root of the Laplacian in R^4n, introduced by the formula D = ∑j=0^3=0 Kjδzj with Kj being the basis of B, and δzj denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra B×R0,4n whose definition involves a delicate construction of the bicomplex Witt basis. The introduction of the operator D can also overturn the prevailing opinion in the Hermitian Clifford analysis in the complex or quaternionic setting that the complex or quaternionic Hermitiean monogenic functions are described by a system of equations instead of by a single equation like classical monogenic functions which are null solutions of Dirac operator. In contrast to the Hermitian Clifford analysis in quaternionic setting, the Poisson brackets of the twisted real Clifford vectors do not vanish in general in the bicomplex setting. For the operator D, we establish the Cauchy integral formula, which generalizes the Martinelli-Bochner formula in the theory of several complex variables.展开更多
A Hilbert transform for H61der continuous circulant (2 × 2) matrix functions, on the d- summable (or fractal) boundary F of a Jordan domain Ω in R2n, has recently been introduced within the framework of Herm...A Hilbert transform for H61der continuous circulant (2 × 2) matrix functions, on the d- summable (or fractal) boundary F of a Jordan domain Ω in R2n, has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the HSlder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the HSlder exponents, the diameter of F and a specific d-sum (d 〉 d) of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary.展开更多
Firstly, the Riemann boundary value problem for a kind of degenerate elliptic sys- tem of the first order equations in R4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Cliffor...Firstly, the Riemann boundary value problem for a kind of degenerate elliptic sys- tem of the first order equations in R4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system's solution, the boundary value problem as stated above is trans- formed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R4 are derived.展开更多
By the characterization of the matrix Hilbert transform in the Hermitian Clifford analysis, we introduce the matrix Szeg5 projection operator for the Hardy space of Hermitean monogenic functions defined on a bounded s...By the characterization of the matrix Hilbert transform in the Hermitian Clifford analysis, we introduce the matrix Szeg5 projection operator for the Hardy space of Hermitean monogenic functions defined on a bounded sub-domain of even dimensional Euclidean space, establish the Kerzman-Stein formula which closely connects the matrix Szego projection operator with the Hardy projection operator onto the Hardy space, and get the matrix Szego projection operator in terms of the Hardy projection operator and its adjoint. Furthermore, we construct the explicit matrix Szego kernel function for the Hardy space on the sphere as an example, and get the solution to a boundary value problem for matrix functions.展开更多
The Mehler formula for the Hermite polynomials allows for an integral representation of the one-dimensional Fractional Fourier transform. In this paper, we introduce a multi-dimensional Fractional Fourier transform in...The Mehler formula for the Hermite polynomials allows for an integral representation of the one-dimensional Fractional Fourier transform. In this paper, we introduce a multi-dimensional Fractional Fourier transform in the framework of Clifford analysis. By showing that it coincides with the classical tensorial approach we are able to prove Mehler's formula for the generalized Clifford-Hermite polynomials of Clifford analysis.展开更多
We study the windowed Fourier transform in the framework of Clifford analysis, which we call the Clifford windowed Fourier transform (CWFT). Based on the spectral representation of the Clifford Fourier transform (...We study the windowed Fourier transform in the framework of Clifford analysis, which we call the Clifford windowed Fourier transform (CWFT). Based on the spectral representation of the Clifford Fourier transform (CFT), we derive several important properties such as shift, modulation, reconstruction formula, orthogonality relation, isometry, and reproducing kernel. We also present an example to show the differences between the classical windowed Fourier transform (WFT) and the CWFT. Finally, as an application we establish a Heisenberg type uncertainty principle for the CWFT.展开更多
Almansi-type decomposition theorem for bi-k-regular functions defined in a star-like domainΩ⊆R^(n+1)×R^(n+1)centered at the origin with values in the Clifford algebra Cl_(2n+2,0)(R)is proved.As a corollary,Alman...Almansi-type decomposition theorem for bi-k-regular functions defined in a star-like domainΩ⊆R^(n+1)×R^(n+1)centered at the origin with values in the Clifford algebra Cl_(2n+2,0)(R)is proved.As a corollary,Almansi-type decomposition theorem for biharmonic functions of degree k is given.展开更多
In the present paper,by extending some fractional calculus to the framework of Clifford analysis,new classes of wavelet functions are presented.Firstly,some classes of monogenic polynomials are provided based on 2-par...In the present paper,by extending some fractional calculus to the framework of Clifford analysis,new classes of wavelet functions are presented.Firstly,some classes of monogenic polynomials are provided based on 2-parameters weight functions which extend the classical Jacobi ones in the context of Clifford analysis.The discovered polynomial sets are next applied to introduce new wavelet functions.Reconstruction formula as well as Fourier-Plancherel rules have been proved.The main tool reposes on the extension of fractional derivatives,fractional integrals and fractional Fourier transforms to Clifford analysis.展开更多
By the Plemelj formula and the compressed fixed point theorem,this paper discusses a kind of boundary value problem for hypermonogenic function vectors in Clifford analysis.And the paper proves the existence and uniqu...By the Plemelj formula and the compressed fixed point theorem,this paper discusses a kind of boundary value problem for hypermonogenic function vectors in Clifford analysis.And the paper proves the existence and uniqueness of the solution to the boundary value problem for hypermonogenic function vectors in Clifford analysis.展开更多
In the first part of this paper, we discuss the Holder continuity of the cauchy integral operator for regular functions and the relation between ‖T{f}‖α and ‖f‖α. In the second part of this paper, we introduce t...In the first part of this paper, we discuss the Holder continuity of the cauchy integral operator for regular functions and the relation between ‖T{f}‖α and ‖f‖α. In the second part of this paper, we introduce the modified cauchy integral operator T^- for regular functions. Firstly, we prove that the operator T^- has a unique fixed point by the Banach's Contraction Mapping Principle. Secondly, we give the Mann iterative sequence, and then we show the iterative sequence strongly converges to the fixed point of the operator T^-.展开更多
In this paper, the Dirac operator on the Klein model for the hyperbolic space is considered. A function space containing L2-functions on the sphere S^m-1 in R^m, which are boundary values of solutions for this operato...In this paper, the Dirac operator on the Klein model for the hyperbolic space is considered. A function space containing L2-functions on the sphere S^m-1 in R^m, which are boundary values of solutions for this operator, is defined, and it is proved that this gives rise to a Hilbert module with a reproducing kernel.展开更多
The paper provides integral representations for solutions to a certain first order partial differential equation natural arising in the factorization of the Lamé–Navier system with the help of Clifford analysis ...The paper provides integral representations for solutions to a certain first order partial differential equation natural arising in the factorization of the Lamé–Navier system with the help of Clifford analysis techniques.These representations look like in spirit to the Borel–Pompeiu and Cauchy integral formulas both in three and higher dimensional setting.展开更多
We introduce two new linear differential operators which are invariant with respect to the unitary group SU(n). They constitute analogues of the twistor and the Rarita-Schwinger operator in the orthogonal case. The ...We introduce two new linear differential operators which are invariant with respect to the unitary group SU(n). They constitute analogues of the twistor and the Rarita-Schwinger operator in the orthogonal case. The natural setting for doing this is Hermitian Clifford Analysis. Such operators are constructed by twisting the two versions of the Hermitian Dirac operator 6z_ and 6z_ and then projecting on irreducible modules for the unitary group. We then study some properties of their spaces of nullsolutions and we find a formulation of the Hermitian Rarita-Schwinger operators in terms of Hermitian monogenic polynomials.展开更多
基金supported by the NSF of Hebei Province(A2022208007)the NSF of China(11571089,11871191)the NSF of Henan Province(222300420397)。
文摘Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quantum mechanics and value problems.In this paper,we first give the definition of a quasi-Cauchy type integral in complex Clifford analysis,and get the Plemelj formula for it.Second,we discuss the H?lder continuity for the Cauchy-type integral operators with values in a complex Clifford algebra.Finally,we prove the existence of solutions for a class of linear boundary value problems and give the integral representation for the solution.
基金Supported by the National Natural Science Foundation of China (10471107)
文摘Under the foundation of Cauchy integral formula on certain distinguished boundary for functions with values in universal Clifford algebra, we define the Cauchy type integral with values in a universal Clifford algebra, obtain its Cauchy principal value and Plemelj formula on certain distinguished boundary.
基金supported by NNSF of China(11171260)RFDP of Higher Education of China(20100141110054)
文摘In this article, we establish the Gauss Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic method. The Cauchy-Pompeiu formula for Clifford-valued functions under the weak condition will be derived as their simple application. Furthermore, Cauchy formula for monogenic functions under the weak condition is derived directly from the Cauchy-Pompeiu formula.
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.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 11371337) and the Research Fund for the Doctoral Program of Higher Education (China) (Grant No. 20123402110068).
文摘Complex Hermitian Clifford analysis emerged recently as a refinement of the theory of several complex variables, while at the same time, the theory of bicomplex numbers motivated by the bicomplex version of quantum mechanics is also under full development. This stimulates us to combine the Hermitian Clifford analysis with the theory of bicomplex number so as to set up the theory of bicomplex Hermitian Clifford analysis. In parallel with the Euclidean Clifford analysis, the bicomplex Hermitian Clifford analysis is centered around the bicomplex Hermitian Dirac operator D: C^∞(R^4n W4n) →4 C^∞(R^4n, W4n), where W4n is the tensor product of three algebras, i.e., the hyperbolic quaternion B, the bicomplex number B, and the Clifford algebra R0,4n. The operator D is a square root of the Laplacian in R^4n, introduced by the formula D = ∑j=0^3=0 Kjδzj with Kj being the basis of B, and δzj denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra B×R0,4n whose definition involves a delicate construction of the bicomplex Witt basis. The introduction of the operator D can also overturn the prevailing opinion in the Hermitian Clifford analysis in the complex or quaternionic setting that the complex or quaternionic Hermitiean monogenic functions are described by a system of equations instead of by a single equation like classical monogenic functions which are null solutions of Dirac operator. In contrast to the Hermitian Clifford analysis in quaternionic setting, the Poisson brackets of the twisted real Clifford vectors do not vanish in general in the bicomplex setting. For the operator D, we establish the Cauchy integral formula, which generalizes the Martinelli-Bochner formula in the theory of several complex variables.
文摘A Hilbert transform for H61der continuous circulant (2 × 2) matrix functions, on the d- summable (or fractal) boundary F of a Jordan domain Ω in R2n, has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the HSlder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the HSlder exponents, the diameter of F and a specific d-sum (d 〉 d) of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary.
基金Supported by the National Science Foundation of China(11401162,11571089,11401159,11301136)the Natural Science Foundation of Hebei Province(A2015205012,A2016205218,A2014205069,A2014208158)Hebei Normal University Dr.Fund(L2015B03)
文摘Firstly, the Riemann boundary value problem for a kind of degenerate elliptic sys- tem of the first order equations in R4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system's solution, the boundary value problem as stated above is trans- formed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R4 are derived.
基金supported by Portuguese funds through the CIDMA Center for Research and Development in Mathematics and Applicationsthe Portuguese Foundation for Science and Technology(FCT–Fundao para a Ciência e a Tecnologia)within project UID/MAT/04106/2013the recipient of a Postdoctoral Foundation from FCT under Grant No. SFRH/BPD/74581/2010
文摘By the characterization of the matrix Hilbert transform in the Hermitian Clifford analysis, we introduce the matrix Szeg5 projection operator for the Hardy space of Hermitean monogenic functions defined on a bounded sub-domain of even dimensional Euclidean space, establish the Kerzman-Stein formula which closely connects the matrix Szego projection operator with the Hardy projection operator onto the Hardy space, and get the matrix Szego projection operator in terms of the Hardy projection operator and its adjoint. Furthermore, we construct the explicit matrix Szego kernel function for the Hardy space on the sphere as an example, and get the solution to a boundary value problem for matrix functions.
基金The work is supported by Research Grant of the University of Macao No.RG021/03-045/QT/FST
文摘The Mehler formula for the Hermite polynomials allows for an integral representation of the one-dimensional Fractional Fourier transform. In this paper, we introduce a multi-dimensional Fractional Fourier transform in the framework of Clifford analysis. By showing that it coincides with the classical tensorial approach we are able to prove Mehler's formula for the generalized Clifford-Hermite polynomials of Clifford analysis.
文摘We study the windowed Fourier transform in the framework of Clifford analysis, which we call the Clifford windowed Fourier transform (CWFT). Based on the spectral representation of the Clifford Fourier transform (CFT), we derive several important properties such as shift, modulation, reconstruction formula, orthogonality relation, isometry, and reproducing kernel. We also present an example to show the differences between the classical windowed Fourier transform (WFT) and the CWFT. Finally, as an application we establish a Heisenberg type uncertainty principle for the CWFT.
基金supported by the National Natural Science Foundation of China(No.11871191)the Science Foundation of Hebei Province(No.A2019106037)+1 种基金the Graduate Student Innovation Project Foundation of Hebei Province(No.CXZZBS2022066)the Key Foundation of Hebei Normal University(Nos.L2018Z01,L2021Z01)
文摘Almansi-type decomposition theorem for bi-k-regular functions defined in a star-like domainΩ⊆R^(n+1)×R^(n+1)centered at the origin with values in the Clifford algebra Cl_(2n+2,0)(R)is proved.As a corollary,Almansi-type decomposition theorem for biharmonic functions of degree k is given.
文摘In the present paper,by extending some fractional calculus to the framework of Clifford analysis,new classes of wavelet functions are presented.Firstly,some classes of monogenic polynomials are provided based on 2-parameters weight functions which extend the classical Jacobi ones in the context of Clifford analysis.The discovered polynomial sets are next applied to introduce new wavelet functions.Reconstruction formula as well as Fourier-Plancherel rules have been proved.The main tool reposes on the extension of fractional derivatives,fractional integrals and fractional Fourier transforms to Clifford analysis.
基金Supported by the National Natural Science Foundation of China (Grant No.10801043)the Natural Science Foundation of Hebei Province (Grant No.A2010000346)the Foundation of Hebei Normal University (GrantNo.L200902)
文摘By the Plemelj formula and the compressed fixed point theorem,this paper discusses a kind of boundary value problem for hypermonogenic function vectors in Clifford analysis.And the paper proves the existence and uniqueness of the solution to the boundary value problem for hypermonogenic function vectors in Clifford analysis.
基金the National Natural Science Foundation of China (No. 10771049 10771050)+1 种基金 the Natural Science Foundation of Hebei Province (No. A2007000225) and the Foundation of Hebei Normal University (No. L2007Q05) the 11th Five-Year Plan Educational and Scientific Issues of Hebei Province (No. O8020147).
文摘In the first part of this paper, we discuss the Holder continuity of the cauchy integral operator for regular functions and the relation between ‖T{f}‖α and ‖f‖α. In the second part of this paper, we introduce the modified cauchy integral operator T^- for regular functions. Firstly, we prove that the operator T^- has a unique fixed point by the Banach's Contraction Mapping Principle. Secondly, we give the Mann iterative sequence, and then we show the iterative sequence strongly converges to the fixed point of the operator T^-.
文摘In this paper, the Dirac operator on the Klein model for the hyperbolic space is considered. A function space containing L2-functions on the sphere S^m-1 in R^m, which are boundary values of solutions for this operator, is defined, and it is proved that this gives rise to a Hilbert module with a reproducing kernel.
基金Supported in part by a grant from Agencia Estatal de Investigacin(PID2019-106433GB-I00/AEI/10.13039/501100011033)Spainpartially supported by Instituto Politécnico Nacional in the framework of SIP programs(SIP20180225)。
文摘The paper provides integral representations for solutions to a certain first order partial differential equation natural arising in the factorization of the Lamé–Navier system with the help of Clifford analysis techniques.These representations look like in spirit to the Borel–Pompeiu and Cauchy integral formulas both in three and higher dimensional setting.
基金sponsored by the relevant grantssupported by the F.W.O. Vlaanderen (Belgium)
文摘We introduce two new linear differential operators which are invariant with respect to the unitary group SU(n). They constitute analogues of the twistor and the Rarita-Schwinger operator in the orthogonal case. The natural setting for doing this is Hermitian Clifford Analysis. Such operators are constructed by twisting the two versions of the Hermitian Dirac operator 6z_ and 6z_ and then projecting on irreducible modules for the unitary group. We then study some properties of their spaces of nullsolutions and we find a formulation of the Hermitian Rarita-Schwinger operators in terms of Hermitian monogenic polynomials.