Perturbation of Frames

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


In applications where bases appear, the question of stability plays an important role. That is, if {f k } k=1 is a basis and {g k } k=1 is in some sense “close” to {f k } k=1 does it follows that {g k } k=1 is also a basis? A classical result states that if {f k } k=1 is a basis for a Banach space X, then a sequence {g k } k=1 in X is also a basis if there exists a constant λ ∈]0,1[ such that for all finite sequences of scalars {c k }. The result is usually attributed to Paley and Wiener [231], but it can be traced back to Neumann [226]: in fact, it is an almost immediate consequence of Theorem A.5.3 with Uf k := g k .


Orthonormal Basis Finite Sequence Haar Wavelet Riesz Basis Tight Frame 
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Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Ole Christensen
    • 1
  1. 1.Department of MathematicsTechnical University of DenmarkLyngbyDenmark

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