POINTWISE CONVERGENCE FOR EXPANSIONS IN SPHERICAL MONOGENICS |
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Authors: | Fei Minggang Qian Tao |
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Institution: | aSchool of Applied Mathematics, University of Electronic Science and Technology of China, Changdu 610054, China;bDepartment of Mathematics, Faculty of Science and Technology, University of Macau, Macao, China |
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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. |
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Keywords: | spherical monogenics pointwise convergence of Fourier-Laplace series generalized Cauchy-Riemann operator unit sphere generalization of Fueter's theorem2000 MR Subject Classification: 42B05 30G35 |
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