Introduction 1 1.1. Schur analysis 2 1.2. Negative squares 2 1.3. The slice hyperholomorphic case 3 2. Preliminaries 4 2.1. Negative squares and kernels 4 2.2. Slice hyperholomorphic functions 6 3. Slice hyperholomorphic operator-valued functions 8 3.1. Realizations 13 4. The Hardy space of the half-space H + 14 5. The Schauder-Tychonoff fixed point theorem 21 5.1. The Schauder-Tychonoff fixed point theorem 21 5.2. An invariant subspace theorem 23 6. The spaces P(S) 24 7. Realization for elements in S κ (J 1 , J 2 ) 26 8. The space L(Φ) and realizations for generalized positive functions 33 8.1. The indefinite case 33 8.2. The positive case 40 References 42

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We study infinite products of reproducing kernels with view to their use in dynamics (of iterated function systems), in harmonic analysis, and in stochastic processes. On the way, we construct a new family of representations of the Cuntz relations. Then, using these representations we associate a fixed filled Julia set with a Hilbert space. This is based on analysis and conformal geometry of a fixed rational mapping R in one complex variable, and its iterations.

We introduce connections between the Cuntz relations and the Hardy space H 2 of the open unit disk D. We then use them to solve a new kind of multipoint interpolation problem in H 2 , where for instance, only a linear combination of the values of a function at given points is preassigned, rather than the values at the points themselves.1991 Mathematics Subject Classification. 42C40, 47A57, 93B28. Key words and phrases. Cuntz relations, Leech's theorem, Schur analysis. D. Alpay thanks the Earl Katz family for endowing the chair which supported his research. The work was done in part while the second named author visited Department of Mathematics, Ben Gurion University of the Negev, supported by a BGU distinguished visiting scientist program. Support and hospitality is much appreciated. We acknowledge discussions with colleagues there, and in the US, Dorin Dutkay, Myung-Sin Song, and Erin Pearse.

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.