关于W_2~1[a,b]空间的两点注记

Two notes on reproducing kernel space W_2~1 [a,b]

讨论再生核空间W12 [a,b]定义中的条件是否可以减弱的问题,得到下面的两个结论:(1)条件u(x)是[a,b]上实的连续函数且u′(x)∈L2 [a,b]不能推出u(x)是[a,b]上实的绝对连续函数; (2)再生核空间W12 [a,b]定义中的条件改为u(x)在[a,b]是连续函数或连续囿变函数,那么函数空间不再是再生核空间.

The authors discuss the problem whether the condition in the definition of a reproducing kernel space can be weakened or not and obtaine two conclusions as follows:(1) The conditions, u(x) being a real continuous function in interval [a,b] and u′(x)∈L^2[a,b], can not deduce the conclusion that u(x) is an absolute continuous function in interval [a,b].(2) If we replace the condition in the definition of a reproducing kernel space W^1_2 [a,b] by the condition of u(x) being continuous or continuous and bounded variation, the functional space would not be a reproducing kernel space.