精确华林不等式的一个推广

An extension of the sharp Wirtinger inequality

给出了一类精确的华林不等式:设a≤x1<x2<…<xr≤b,则对任给满足条件f(x1)=f(x2)=…=f(xr)=0的函数f∈Wq^(r)[a,b],有||f||_(p)≤C(p,q)(b-a)^(r+1/p-1/q)||f^((r))||_(q),1≤p,q≤∞.首先,基于拉格朗日插值的积分型余项公式,将C(p,q)的计算转化为一个积分算子的范数;其次,将C(1,1)和C(∞,∞)的值转化为2个显式积分表达式,并将C(2,2)的值转化为计算一个希尔伯特-施密特算子的最大特征值;最后,用一个例子说明.

In this study,we give a sharp Wirtinger inequality||f||_(p)≤C(p,q)(b-a)^(r+1/p-1/q)||f^((r))||_(q),1≤p,q≤∞.For an arbitrary f∈Wq^(r)[a,b]with f(x1)=f(x2)=…=f(xr)=0,a≤x1<x2<…<xr≤b.From integral type remainder of Lagrange interpolation,we refer computation of C(p,q)to the norm of an integral operator.We refer values of C(1,1)and C(∞,∞)to two explicit integral expressions and value of C(2,2)to computation of maximum eigenvalue of a Hilbert-Schmidt operator.An example was then given to show our method.