Sobolev空间H^s(R^d)上波包系的框架性质

Frame Properties of Wave Packet Systems in Sobolev Spaces H^s(R^d)

波包系是通过对有限个函数做伸缩、平移和调制三种运算生成的一种新型函数系,因此传统的小波系和Gabor系都是它的特殊情况.该文首先给出了Sobolev空间H^s(R^d)中一个广义平移函数系成为Bessel点列或框架的充分条件,然后结合波包系是一类特殊的广义平移函数系这一结果,给出了高维Sobolev空间H^s(R^d)上波包系成为框架的一个充分条件.最后,利用矩阵的特征值理论,该文证明了:如果函数g的Fourier变换在某一开球中大于某个正数,那么由它生成的波包系不能成为H^s(R^d)的一个框架.

The traditional wave systems and Gabor systems are special cases of wave packet systems,which are obtained by applying a combination of dilations,translations and modulations to a finite family of functions.In this paper,first,a sufficient condition for a generalized shift-inveriant system to be a Bessel sequence or even a frame for the Sobolev spaces H-s（R-d） is established.Secondly,combining with the result that the wave packet system is a special case of the generalized shift-invariant system,the sufficient condition for the wave packet system to be a frame for the Sobolev spaces H-s（R-d） is obtained.Finally,using the eigenvalues of the matrix theory,this paper proves that if the Fourier transform of the function g on a certain open ball is greater than some positive number,then the wave packet system which is generated by it,cannot to be a frame for H-s（R-d）.