闭区间上Zygmund函数的延拓定理

On the Extension Theorem for Zygmund Functions in Closed Interval

设f(x)是闭区间I上的连续函数,f(x)为I上的Zygmund函数．如果存在常数C≥0,使得f(x)满足|f(x+t)-2f(x)+f(x-t)|<Ct,则对一切x,x±t∈I,t>0成立．可将其延拓成上的Zygmund函数的充分条件,并估计其范数‖f‖z．

Let f（x） be continuous on closed interval I, f（x） is called a Zygmund function on I if there exists one constant C≥0 such that J｜f（x＋t）-2f（x）＋f（x）｜〈Ct for all x,x±t∈I,t〉0. We give sufficient condition for such Zygmund function which can be extended to a Zygmund function on R and give estimation for its norm ‖f‖.