As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the researc...As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.展开更多
Let S^(d-1) = {x : |x| = 1} be a unit sphere of the d-dimensional Euclideanspace R^d and let H^p = H^p(S^(d-1)) (0 < p ≤ 1) denote the real Hardy space on S^(d-1). For 0 < p≤ 1 and f ∈ H^p(S^(d-1)), let E_j (...Let S^(d-1) = {x : |x| = 1} be a unit sphere of the d-dimensional Euclideanspace R^d and let H^p = H^p(S^(d-1)) (0 < p ≤ 1) denote the real Hardy space on S^(d-1). For 0 < p≤ 1 and f ∈ H^p(S^(d-1)), let E_j (f, H^p) (j =0,1,...) be the best approximation of f byspherical polynomials of degree less than or equal to j, in the space H^p(S^(d-1)). Given adistribution f on S^(d-1), its Cesaro mean of order δ > -1 is denoted by σ_k~δ(f). For 0 < p ≤1, it is known that δ(p) := (d-1)/p - d/2 is the critical index for the uniform summability ofσ_k~δ(f) in the metric H^p.展开更多
基金Supported by the NSF of China under the Grant 10471010partially by the NSERC Canada under Grant G121211001
文摘As early as in 1990, Professor Sun Yongsheng, suggested his students at Beijing Normal University to consider research problems on the unit sphere. Under his guidance and encouragement his students started the research on spherical harmonic analysis and approximation. In this paper, we incompletely introduce the main achievements in this area obtained by our group and relative researchers during recent 5 years （2001-2005）. The main topics are： convergence of Cesaro summability, a.e. and strong summability of Fourier-Laplace series; smoothness and K-functionals; Kolmogorov and linear widths.
基金The authors are partially supported by NNSF of China under the grant#10071007
文摘Let S^(d-1) = {x : |x| = 1} be a unit sphere of the d-dimensional Euclideanspace R^d and let H^p = H^p(S^(d-1)) (0 < p ≤ 1) denote the real Hardy space on S^(d-1). For 0 < p≤ 1 and f ∈ H^p(S^(d-1)), let E_j (f, H^p) (j =0,1,...) be the best approximation of f byspherical polynomials of degree less than or equal to j, in the space H^p(S^(d-1)). Given adistribution f on S^(d-1), its Cesaro mean of order δ > -1 is denoted by σ_k~δ(f). For 0 < p ≤1, it is known that δ(p) := (d-1)/p - d/2 is the critical index for the uniform summability ofσ_k~δ(f) in the metric H^p.
