该文研究Dirichlet及随机Dirichlet级数在水平直线或半直线上的增长性,包含关于Taylor级数的相应结果,例如下列简单结果:设Taylor级数F_(z)=sum from n=0 to ∞有收敛半径∞或1,其中0=μ_0<μ_n↑,μ_n∈N,sum from(1/μ_n)<∞....该文研究Dirichlet及随机Dirichlet级数在水平直线或半直线上的增长性,包含关于Taylor级数的相应结果,例如下列简单结果:设Taylor级数F_(z)=sum from n=0 to ∞有收敛半径∞或1,其中0=μ_0<μ_n↑,μ_n∈N,sum from(1/μ_n)<∞.如果这级数有级ρ(在收敛半径是∞或1时,“级”的意义不同),那么在第一种情形。它在从原点出发的每条射线上有级p;在第二种情形,在单位圆盘的每条射线上有级ρ．展开更多
Chaundy and Jolliffe proved that if {a n } is a non-increasing (monotonic) real sequence with lim n →∞ a n = 0, then a necessary and sufficient condition for the uniform convergence of the series ∑∞ n=1 a n sin nx...Chaundy and Jolliffe proved that if {a n } is a non-increasing (monotonic) real sequence with lim n →∞ a n = 0, then a necessary and sufficient condition for the uniform convergence of the series ∑∞ n=1 a n sin nx is lim n →∞ na n = 0. We generalize (or weaken) the monotonic condition on the coefficient sequence {a n } in this classical result to the so-called mean value bounded variation condition and prove that the generalized condition cannot be weakened further. We also establish an analogue to the generalized Chaundy-Jolliffe theorem in the complex space.展开更多
文摘该文研究Dirichlet及随机Dirichlet级数在水平直线或半直线上的增长性,包含关于Taylor级数的相应结果,例如下列简单结果:设Taylor级数F_(z)=sum from n=0 to ∞有收敛半径∞或1,其中0=μ_0<μ_n↑,μ_n∈N,sum from(1/μ_n)<∞.如果这级数有级ρ(在收敛半径是∞或1时,“级”的意义不同),那么在第一种情形。它在从原点出发的每条射线上有级p;在第二种情形,在单位圆盘的每条射线上有级ρ．
基金supported by National Sciences and Engineering Research Council of CanadaNational Natural Science Foundation of China (Grant No. 10471130)
文摘Chaundy and Jolliffe proved that if {a n } is a non-increasing (monotonic) real sequence with lim n →∞ a n = 0, then a necessary and sufficient condition for the uniform convergence of the series ∑∞ n=1 a n sin nx is lim n →∞ na n = 0. We generalize (or weaken) the monotonic condition on the coefficient sequence {a n } in this classical result to the so-called mean value bounded variation condition and prove that the generalized condition cannot be weakened further. We also establish an analogue to the generalized Chaundy-Jolliffe theorem in the complex space.
