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