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Reproducing kernels for polynomial null-solutions of Dirac operators

(2016) CONSTRUCTIVE APPROXIMATION. 44(3). p.339-383
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Abstract
It is well known that the reproducing kernel of the space of spherical harmonics of fixed homogeneity is given by a Gegenbauer polynomial. By going over to complex variables and restricting to suitable bihomogeneous subspaces, one obtains a reproducing kernel expressed as a Jacobi polynomial, which leads to Koornwinder's celebrated result on the addition formula. In the present paper, the space of Hermitian monogenics, which is the space of polynomial bihomogeneous null-solutions of a set of two complex conjugated Dirac operators, is considered. The reproducing kernel for this space is obtained and expressed in terms of sums of Jacobi polynomials. This is achieved through use of the underlying Lie superalgebra , combined with the equivalence between the inner product on the unit sphere and the Fischer inner product. The latter also leads to a new proof in the standard Dirac case related to the Lie superalgebra osp (1 vertical bar 2).
Keywords
Clifford analysis, Hermitian Clifford analysis, Dirac operator, Spherical harmonics, Jacobi polynomials, Gegenbauer polynomials, Reproducing kernels

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Chicago
De Bie, Hendrik, Franciscus Sommen, and Michael Wutzig. 2016. “Reproducing Kernels for Polynomial Null-solutions of Dirac Operators.” Constructive Approximation 44 (3): 339–383.
APA
De Bie, H., Sommen, F., & Wutzig, M. (2016). Reproducing kernels for polynomial null-solutions of Dirac operators. CONSTRUCTIVE APPROXIMATION, 44(3), 339–383.
Vancouver
1.
De Bie H, Sommen F, Wutzig M. Reproducing kernels for polynomial null-solutions of Dirac operators. CONSTRUCTIVE APPROXIMATION. 2016;44(3):339–83.
MLA
De Bie, Hendrik, Franciscus Sommen, and Michael Wutzig. “Reproducing Kernels for Polynomial Null-solutions of Dirac Operators.” CONSTRUCTIVE APPROXIMATION 44.3 (2016): 339–383. Print.
@article{7009202,
  abstract     = {It is well known that the reproducing kernel of the space of spherical harmonics of fixed homogeneity is given by a Gegenbauer polynomial. By going over to complex variables and restricting to suitable bihomogeneous subspaces, one obtains a reproducing kernel expressed as a Jacobi polynomial, which leads to Koornwinder's celebrated result on the addition formula. In the present paper, the space of Hermitian monogenics, which is the space of polynomial bihomogeneous null-solutions of a set of two complex conjugated Dirac operators, is considered. The reproducing kernel for this space is obtained and expressed in terms of sums of Jacobi polynomials. This is achieved through use of the underlying Lie superalgebra , combined with the equivalence between the inner product on the unit sphere and the Fischer inner product. The latter also leads to a new proof in the standard Dirac case related to the Lie superalgebra osp (1 vertical bar 2).},
  author       = {De Bie, Hendrik and Sommen, Franciscus and Wutzig, Michael},
  issn         = {0176-4276},
  journal      = {CONSTRUCTIVE APPROXIMATION},
  language     = {eng},
  number       = {3},
  pages        = {339--383},
  title        = {Reproducing kernels for polynomial null-solutions of Dirac operators},
  url          = {http://dx.doi.org/10.1007/s00365-016-9326-6},
  volume       = {44},
  year         = {2016},
}

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