Sobolev空间Hs(Rd)上波包系的框架性质

数学物理学报 ›› 2016, Vol. 36 ›› Issue (5): 848-860.

Sobolev空间H^s(R^d)上波包系的框架性质

徐美玉, 鲁大勇

河南大学数学与统计学院河南开封 475001

收稿日期:2015-11-14 修回日期:2016-03-24 出版日期:2016-10-26 发布日期:2016-10-26
通讯作者: 鲁大勇,E-mail:dayonglu@163.com E-mail:dayonglu@163.com
作者简介:徐美玉,E-mail:xumeiyu604@163.com;鲁大勇,E-mail:dayonglu@163.com
基金资助:
国家自然科学基金（61471410）和河南省教育厅科学技术研究重点项目（13A110072）资助

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

Xu Meiyu, Lu Dayong

School of Mathematics and Statistics, Henan University, Henan Kaifeng

Received:2015-11-14 Revised:2016-03-24 Online:2016-10-26 Published:2016-10-26
Supported by:
Supported by the NSFC (61471410) and the Key Project of Science and Technology Research of Education Department of Henan Province (13A110072)

摘要/Abstract

摘要：

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.

关键词: Sobolev空间, 框架, 波包系, 广义平移不变系

Abstract:

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).

Key words: Sobolev spaces, Frames, Wave packet systems, Generalized shift-invariant systems

中图分类号:

O174.2

徐美玉, 鲁大勇. Sobolev空间H^s(R^d)上波包系的框架性质[J]. 数学物理学报, 2016, 36(5): 848-860.

Xu Meiyu, Lu Dayong. Frame Properties of Wave Packet Systems in Sobolev Spaces H^s(R^d)[J]. Acta mathematica scientia,Series A, 2016, 36(5): 848-860.

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