# Comparison of Discrete and Continuous Wavelet Transforms

**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 *L*^{2}(ℝ^{n}) 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 *L*^{2}(ℝ^{n}) analysis and fractal analysis. To facilitate the discussion of the interplay between discrete (used by engineers) and...

## Keywords

Hilbert Space Wavelet Transform Continuous Wavelet Transform Continuous Wavelet Multiresolution Analysis## Notes

### Acknowledgments

We thank Professors Dorin Dutkay, Gabriel Picioroaga, and Judy Packer for the helpful discussions.

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