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

Authors

Daniel Alpay, Chapman UniversityFollow
Palle Jorgensen, Univeristy of Iowa
Izchak Lewkowicz, Ben Gurion University of the Negev

Document Type

Conference Proceeding

Publication Date

2013

Abstract

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.

Comments

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

Recommended Citation

D. Alpay, P. Jorgensen and I. Lewkowicz. Extending wavelet filters. Infinite dimensions, the non-rational case, and indefinite-inner product spaces. In Excursions in Harmonic Analysis, Volume 2, Proceedings of the February Fourier Talks (FFT 2006 -- 2011), Springer-Birkhauser Applied and Numerical Harmonic Analysis (ANHA) Book Series, (2013), pp..

Copyright

Springer