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Generalized Filters

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Generalized Multiresolution Analyses

Part of the book series: Applied and Numerical Harmonic Analysis ((LN-ANHA))

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Abstract

Filters in a GMRA encode the containments V−1 ⊂ V0 and W−1 = δ−1W0 ⊂ V0 as relationships between scaling functions, wavelet functions, and their dilates. Classical filters were defined in \(L^2(\mathbb R)\) in terms of Fourier transforms of these functions, and were used to build MRA’s and orthonormal wavelets with desirable properties. Generalized filters take advantage of the GMRA structure by using the unitary operator given by spectral multiplicity in place of the Fourier transform.

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Authors and Affiliations

  1. Department of Mathematics, The Colorado College, Colorado Springs, CO, USA

    Kathy D. Merrill

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  1. Kathy D. Merrill
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Merrill, K.D. (2018). Generalized Filters. In: Generalized Multiresolution Analyses. Applied and Numerical Harmonic Analysis(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-99175-7_5

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