Acta mathematica scientia,Series A ›› 2016, Vol. 36 ›› Issue (5): 848-860.

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Frame Properties of Wave Packet Systems in Sobolev Spaces Hs(Rd)

Xu Meiyu, Lu Dayong   

  1. 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)

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 Hs(Rd) 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 Hs(Rd) 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 Hs(Rd).

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

CLC Number: 

  • O174.2
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