L^2情形的Fourier-Laplace级数几乎处处收敛的刻画

A Characterization of A.E. Convergence of Fourier-Laplace Series in L^2

设Ωn 为Rn 中的单位球面 ,f∈L2 (Ωn) ,σ0N(f) (x)为f(x)的Fouier Laplace展开式的部分和 ,wr(f,t) 2 为其r阶连续模 .证明了当∫10wr(f,t) 22t (1+sinlnt)dt <+∞时 ,limN→∞σ0N(f) (x) =f(x) ,a .e .x∈Ωn。

Let Ω n the unit sphere in R n, f∈L 2(Ω n), σ 0 N(f)(x) the partial sums of Fourier-Laplace expansion of f(x) and w r(f,t) 2 r the modulus of smoothness of f(x). It is proved that limN→∞σ 0 N(f)(x)=f(x), a.e.x∈Ω n, if ∫ 1 0w r(f,t) 2 2t(1+sin lnt)dt<+∞. Some well-known results which have be known are improved.