广义光滑模和广义凸性模的性质

Properties of Generalized Modulus of Smoothness and Generalized Modulus of Convexity

基于广义光滑模的定义,研究了Banach空间下的广义光滑模与t之间的关系,证明了一致非方的三个等价条件以及关于广义光滑模的四个等价命题。此外证明了Banach空间和超自反的Banach空间分别满足limt→0*ρX^α*(t)/t<1/2和ρX^α*(t)<α+3/2*t*ω*(x)-1,t·ω*(X)≤1的条件下具有一致正规结构,ρX^α*(t)和ω(X)分别为广义光滑模和弱正交系数。最后给出了Vx,y∈X当‖x‖^2+‖y‖^2=2时关于广义凸性模的一个不等式。

Based on the definitions of generalized modulus of smoothness,the relation between generalized modulus of smoothness and t in the Banach spaces is studied,which proves three equivalent conditions of uniform normal structure and four equivalent propositions of generalized modulus of smoothness.In addition,which is proved that the Banach space and the super reflexive Banach space satisfy conditions of limt→0*ρX^α*(t)/t<1/2andρX^α*(t)<α+3/2*t*ω*(x)-1,t·ω*(X)≤1 have uniform normal structure.ραX(t)andω(X)are generalized modulus of smoothness and weak orthogonal coefficient respectively.Finally,which gives an inequality about the generalized convex modulus when‖x‖^2+‖y‖^2=2,Vx,y∈X.