数学物理学报(英文版) ›› 2009, Vol. 29 ›› Issue (5): 1241-1250.doi: 10.1016/S0252-9602(09)60101-6

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POINTWISE CONVERGENCE FOR EXPANSIONS IN SPHERICAL MONOGENICS

 费铭岗, 钱涛   

  1. 1.School of Applied Mathematics, University of Electronic Science and Technology of China, Changdu 610054, China
    2.Department of Mathematics, Faculty of Science and Technology, University of Macau, Macao, |China
  • 收稿日期:2007-08-23 出版日期:2009-09-20 发布日期:2009-09-20

POINTWISE CONVERGENCE FOR EXPANSIONS IN SPHERICAL MONOGENICS

 FEI Ming-Gang, QIAN Tao   

  1. 1.School of Applied Mathematics, University of Electronic Science and Technology of China, Changdu 610054, China
    2.Department of Mathematics, Faculty of Science and Technology, University of Macau, Macao, |China
  • Received:2007-08-23 Online:2009-09-20 Published:2009-09-20

摘要:

We offer a new approach to deal with the pointwise convergence of Fourier-Laplace 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.

关键词: spherical monogenics, generalized CauchyRiemann operator, unit sphere, generalization of Fueters theorem

Abstract:

We offer a new approach to deal with the pointwise convergence of Fourier-Laplace 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.

中图分类号: 

  • 42B05