Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Comparison of Discrete and Continuous Wavelet Transforms

Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_77-2

Our purpose is to outline a number of direct links between the two cases of wavelet analysis: continuous and discrete. The theme of the first is perhaps best known, for example, the creation of compactly supported wavelets in L2(ℝn) with suitable properties such as localization, vanishing moments, and differentiability. The second (discrete) deals with computation, with sparse matrices, and with algorithms for encoding digitized information such as speech and images. This is centered on constructive approaches to subdivision filters, their matrix representation (by sparse matrices), and corresponding fast algorithms. For both approaches, we outline computational transforms; but our emphasis is on effective and direct links between computational analysis of discrete filters on the one side and on continuous wavelets on the other. By the latter, we include both L2(ℝn) analysis and fractal analysis. To facilitate the discussion of the interplay between discrete (used by engineers) and...


Hilbert Space Wavelet Transform Continuous Wavelet Transform Continuous Wavelet Multiresolution Analysis 
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We thank Professors Dorin Dutkay, Gabriel Picioroaga, and Judy Packer for the helpful discussions.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsThe University of IowaIowa CityUSA
  2. 2.Department of Mathematics and StatisticsSouthern Illinois UniversityEdwardsvilleUSA