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System theory and orthogonal multi-wavelets

Lookup NU author(s): Professor Mihai Putinar

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Abstract

In this paper we provide a complete and unifying characterization of compactly supported univariate scalar orthogonal wavelets and vector-valued or matrixvalued orthogonal multi-wavelets. This characterization is based on classical results from system theory and basic linear algebra. In particular, we show that the corresponding wavelet and multi-wavelet masks are identified with a transfer function F (z ) = A + Bz (I − Dz ) 􀀀1 C, z ∈ D = {z ∈ C : |z| < 1}, of a conservative linear system. The complex matrices A, B, C, D define a block circulant unitary matrix. Our results show that there are no intrinsic differences between the elegant wavelet construction by Daubechies or any other construction of vector-valued or matrix-valued multi-wavelets. The structure of the unitary matrix defined by A, B, C, D allows us to parametrize in a systematic way all classes of possible wavelet and multi-wavelet masks together with the masks of the corresponding refinable functions.

Publication metadata

Author(s): Charina M, Conti C, Cotronei M, Putinar M

Publication type: Article

Publication status: Published

Journal: Journal of Approximation Theory

Year: 2017

Volume: 238

Pages: 85-102

Print publication date: 01/02/2019

Online publication date: 05/10/2017

Acceptance date: 06/09/2017

Date deposited: 21/06/2017

ISSN (print): 0021-9045

ISSN (electronic): 1096-0430

Publisher: Academic Press

URL: https://doi.org/10.1016/j.jat.2017.09.004

DOI: 10.1016/j.jat.2017.09.004

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