Extending Wavelet Filters. Infinite Dimensions, the Non-Rational Case, and Indefinite-Inner Product Spaces

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Conference Proceeding

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In this paper we are discussing various aspects of wavelet filters. While there are earlier studies of these filters as matrix valued functions in wavelets, in signal processing, and in systems, we here expand the framework. Motivated by applications, and by bringing to bear tools from reproducing kernel theory, we point out the role of non-positive definite Hermitian inner products (negative squares), for example Krein spaces, in the study of stability questions. We focus on the nonrational case, and establish new connections with the theory of generalized Schur functions and their associated reproducing kernel Pontryagin spaces, and the Cuntz relations.


This is a pre-copy-editing, author-produced PDF of a conference proceeding published in Proceedings of the February Fourier Talks in 2-13. The definitive publisher-authenticated version is available online at DOI: 10.1007/978-0-8176-8379-5_5