Frames Versus Riesz Bases

  • Ole Christensen
Chapter
First Online:
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We have already seen that Riesz bases are frames. In this chapter we exploit the relationship between these two concepts further. In particular, we give a number of equivalent conditions for a frame to be a Riesz basis.

We have often spoken about a frame in an intuitive sense as some kind of “overcomplete basis.” It turns out that, in the technical sense, one has to be careful with such statements. In fact, we will prove the existence of a frame which has no relation to a basis: no subfamily of the frame forms a basis. On the other hand, sufficient conditions for a frame to contain a Riesz basis as a subfamily are also given.

Keywords

Moment Problem Riesz Basis Tight Frame Gabor Frame Lower Frame 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Ole Christensen
    • 1
  1. 1.Department of Mathematics and Computer ScienceTechnical University of DenmarkLyngbyDenmark

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