New higher dimensional distributions are introduced in the framework of Clifford analysis.They complete the picture already established in previous work, offering unity and structuralclarity. Amongst them are the buil...New higher dimensional distributions are introduced in the framework of Clifford analysis.They complete the picture already established in previous work, offering unity and structuralclarity. Amongst them are the building blocks of the principal value distribution, involvingspherical harmonics, considered by Horvath and Stein.展开更多
Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions, i.e., null solutions to a first order vector valued rotation in- variant differential operator called the Dirac ...Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions, i.e., null solutions to a first order vector valued rotation in- variant differential operator called the Dirac operator. More recently, Hermitian Clifford analysis has emerged as a new branch, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions, called Hermitian monogenic functions, to two Hermitian Dirac operators and which are invariant under the action of the unitary group. In Euclidean Clifford analysis, the Teodorescu operator is the right inverse of the Dirac operator __0. In this paper, Teodorescu operators for the Hermitian Dirac operators c9~_ and 0_~, are constructed. Moreover, the structure of the Euclidean and Hermitian Teodor- escu operators is revealed by analyzing the more subtle behaviour of their components. Finally, the obtained inversion relations are still refined for the differential operators is- suing from the Euclidean and Hermitian Dirac operators by splitting the Clifford algebra product into its dot and wedge parts. Their relationship with several complex variables theory is discussed.展开更多
Around the central theme of 'square root' of the Laplace operator it is shown that the classical Riesz potentials of the first and of the second kind allow for an explicit expression of so-called Hilbert-Dirac...Around the central theme of 'square root' of the Laplace operator it is shown that the classical Riesz potentials of the first and of the second kind allow for an explicit expression of so-called Hilbert-Dirac convolution operators involving natural and complex powers of the Dirac operator.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
文摘New higher dimensional distributions are introduced in the framework of Clifford analysis.They complete the picture already established in previous work, offering unity and structuralclarity. Amongst them are the building blocks of the principal value distribution, involvingspherical harmonics, considered by Horvath and Stein.
文摘Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions, i.e., null solutions to a first order vector valued rotation in- variant differential operator called the Dirac operator. More recently, Hermitian Clifford analysis has emerged as a new branch, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions, called Hermitian monogenic functions, to two Hermitian Dirac operators and which are invariant under the action of the unitary group. In Euclidean Clifford analysis, the Teodorescu operator is the right inverse of the Dirac operator __0. In this paper, Teodorescu operators for the Hermitian Dirac operators c9~_ and 0_~, are constructed. Moreover, the structure of the Euclidean and Hermitian Teodor- escu operators is revealed by analyzing the more subtle behaviour of their components. Finally, the obtained inversion relations are still refined for the differential operators is- suing from the Euclidean and Hermitian Dirac operators by splitting the Clifford algebra product into its dot and wedge parts. Their relationship with several complex variables theory is discussed.
文摘Around the central theme of 'square root' of the Laplace operator it is shown that the classical Riesz potentials of the first and of the second kind allow for an explicit expression of so-called Hilbert-Dirac convolution operators involving natural and complex powers of the Dirac operator.
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.
