Celebratio Mathematica — Freedman

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article M. H. Freedman and W. H. Press: “Truncation of wavelet matrices: Edge effects and the reduction of topological control,” Linear Algebra Appl. 234 (1996), pp. 1–19. MR 1368768 Zbl 0843.65100

Edge effects and Gibbs phenomena are a ubiquitous problem in signal processing. We show how this problem can arise from a mismatch between the “topology” of the data \( D \) (e.g., an interval in the case of a time series or a rectangle in the case of a photographic image) and the topology \( X \) (often a circle or torus) natural to the construction of the transformation \( O \). The notion of a manifold control space \( X \) for an orthogonal transformation \( O \) is introduced. It is proved that no matter how complicated \( X \) is, \( O \) may be “truncated” to an \( O^{\prime} \) with control space \( D \), homeomorphic to an interval or a product of intervals. This yields a new, topologically motivated approach to edge effects. We give the complete details for applying this approach to the discrete Daubechies transform of functions on the unit interval so that no data are wrapped around from one end of the interval to the other.