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Continuity of the Joint Spectral Radius: Application to Wavelets

  • Conference paper
Linear Algebra for Signal Processing

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 69))

Abstract

The joint spectral radius is the extension to two or more matrices of the (ordinary) spectral radius ρ(A) = max ∣λ i (A)∣ = lim ∥A m∥1/m. The extension allows matrix products Π m taken in all orders, so that norms and eigenvalues are difficult to estimate. We show that the limiting process does yield a continuous function of the original matrices—this is their joint spectral radius. Then we describe the construction of wavelets from a dilation equation with coefficients c k . We connect the continuity of those wavelets to the value of the joint spectral radius of two matrices whose entries are formed from the c k .

Partially supported by National Science Foundation Grant DMS-9006220

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Author information

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Skiles Building, Room 242, Atlanta, Georgia, USA, 30332

    Christopher Heil

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA

    Gilbert Strang

Authors
  1. Christopher Heil
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  2. Gilbert Strang
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Editor information

Editors and Affiliations

  1. School of Electrical Engineering 337 E&TC, Cornell University, Ithaca, NY, 14853-3801, USA

    Adam Bojanczyk

  2. Dartmouth College, Thayer School of Engineering, Hanover, NH, 03755-8000, USA

    George Cybenko

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© 1995 Springer-Verlag New York, Inc.

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Heil, C., Strang, G. (1995). Continuity of the Joint Spectral Radius: Application to Wavelets. In: Bojanczyk, A., Cybenko, G. (eds) Linear Algebra for Signal Processing. The IMA Volumes in Mathematics and its Applications, vol 69. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4228-4_4

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  • DOI: https://doi.org/10.1007/978-1-4612-4228-4_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-8703-2

  • Online ISBN: 978-1-4612-4228-4

  • eBook Packages: Springer Book Archive

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