Let ∑<sub>n-1</sub> be the unit sphere in the n-dimensional Euclidean space R<sup>n</sup>.For a function f ∈L(∑<sub>n-1</sub>) denote by σ<sub>N</sub><sup>δ&l...Let ∑<sub>n-1</sub> be the unit sphere in the n-dimensional Euclidean space R<sup>n</sup>.For a function f ∈L(∑<sub>n-1</sub>) denote by σ<sub>N</sub><sup>δ</sup>（f） the Cesàro means of order δ of the Fourier-Laplace series of f.The special value λ∶=（n-2）/2 of δ is known as the critical index.In the case when n is even,this paper proves the existence of the‘rare’sequence {n<sub>k</sub>} such that the summability 1/N sum from k=1 to N σ<sub>n<sub>k</sub></sub><sup>λ</sup>（f）（x）→f（x）,N→∞ takes place at each Lebesgue point satisfying some antipole conditions.展开更多
A covering lemma on the unit sphere is established and then is applied to establish an almost everywhere convergence test of Marcinkiewicz type for the Fourier-Laplace series on the unit sphere which can be stated as ...A covering lemma on the unit sphere is established and then is applied to establish an almost everywhere convergence test of Marcinkiewicz type for the Fourier-Laplace series on the unit sphere which can be stated as follows：Theorem Suppose f ∈ L（En-1）, n≥ 3. If f satisfies the condition1/θ^n-1∫D（x,θ）｜f（y）-f（x）｜dy=O（1/｜logθ｜）,as θ→0＋,at every point x in a set E of positive measure in Σn-1, then the Cesàro means of critical order ,n-2/2 of the Fourier-Laplace series of f converge to f at almost every point x in E.展开更多
基金Project supported by the Natural Science Foundation of China under Grant ≠ 19771009
文摘Let ∑<sub>n-1</sub> be the unit sphere in the n-dimensional Euclidean space R<sup>n</sup>.For a function f ∈L(∑<sub>n-1</sub>) denote by σ<sub>N</sub><sup>δ</sup>（f） the Cesàro means of order δ of the Fourier-Laplace series of f.The special value λ∶=（n-2）/2 of δ is known as the critical index.In the case when n is even,this paper proves the existence of the‘rare’sequence {n<sub>k</sub>} such that the summability 1/N sum from k=1 to N σ<sub>n<sub>k</sub></sub><sup>λ</sup>（f）（x）→f（x）,N→∞ takes place at each Lebesgue point satisfying some antipole conditions.
基金supported by the NSF of China,No.10471010the Natural Science Foundation of Tianjin Normal University,52LJ32The Planned Project of the Development Foundation of Science and Technology of Universities in Tianjin
文摘A covering lemma on the unit sphere is established and then is applied to establish an almost everywhere convergence test of Marcinkiewicz type for the Fourier-Laplace series on the unit sphere which can be stated as follows：Theorem Suppose f ∈ L（En-1）, n≥ 3. If f satisfies the condition1/θ^n-1∫D（x,θ）｜f（y）-f（x）｜dy=O（1/｜logθ｜）,as θ→0＋,at every point x in a set E of positive measure in Σn-1, then the Cesàro means of critical order ,n-2/2 of the Fourier-Laplace series of f converge to f at almost every point x in E.
