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The main result is that forJ = ((1)(0) (0)(-1))every J-unitary 2 x 2-matrix polynomial on the unit circle is an essentially unique product of elementary J-unitary 2 x 2-matrix polynomials which are either of degree 1 or 2k. This is shown by means of the generalized Schur transformation introduced in [Ann. Inst. Fourier 8 (1958) 211; Ann. Acad. Sci. Fenn. Ser. A I 250 (9) (1958) 1-7] and studied in [Pisot and Salem Numbers, Birkhauser Verlag, Basel, 1992; Philips J. Res. 41 (1) (1986) 1-54], and also in the first two parts [Operator Theory: Adv. Appl. 129, Birkhauser Verlag, Basel, 2000, p. 1; Monatshefte fur Mathematik, in press] of this series. The essential tool in this paper are the reproducing kernel Pontryagin spaces associated with generalized Schur functions. (C) 2003 Elsevier Science Inc. All fights reserved.

We prove a realization theorem for rational functions of several complex variables.

We define and study the ring of rational functions in the hyperholomorphic setting. We give a number of equivalent characterizations of rationality. The Cauchy–Kovalevskaya product plays an important role in the arguments. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Resume Nous definissons une theorie des fonctions et un espace de Hardy sur l'arbre dyadique, qui etendent de maniere naturelle la notion de fonction analytique, la formule de Cauchy et l'espace de Hardy du disque unite. Nous definissons l'analogue d'un facteur de Blaschke et demontrons aussi un theoreme d'interpolation homogene dans ce cadre. Pour citer cet article : D. Alpay, D. Volok, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

One proves, using methods of Hilbert spaces with a reproducing kernel, that any bounded analytic function on a complex curve in general position in the unit ball of C$^n$ extends to a function in the Schur class of the ball.

In the first paper of this series (Daniel Alpay, Tomas Azizov, Aad Dijksma, and Heinz Langer: The Schur algorithm for generalized Schur functions I: coisometric realizations, Operator Theory: Advances and Applications 129 (2001), pp. 1-36) it was shown that for a generalized Schur function s(z), which is the characteristic function of a coisometric colligation V with state space being a Pontryagin space, the Schur transformation corresponds to a finite-dimensional reduction of the state space, and a finite-dimensional perturbation and compression of its main operator. In the present paper we show that these formulas can be explained using simple relations between V and the colligation of the reciprocal s(z)(-1) of the characteristic function s(z) and general factorization results for characteristic functions.

Resume Nous demontrons un theoreme de type Gleason pour les fonctions hyper-analytiques dans la boule unite de R 4 . Nous donnons une interpretation du resultat en termes de paires de fonctions definies dans la boule unite de C 2 . Enfin, nous utilisons le theoreme pour etudier le probleme d'interpolation homogene dans le cadre des fonctions hyper-analytiques. Pour citer cet article : D. Alpay, M. Shapiro, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 889–894.

An indefinite generalization of Nudel ′ ’ man’s problem is used in a systematic approach to interpolation theorems for generalized Schur and Nevanlinna functions with interior and boundary data. Besides known results on existence criteria for Pick-Nevanlinna and Carathéodory-Fejér interpolation, the method yields new results on generalized interpolation in the sense of Sarason and boundary interpolation, including properties of the finite Hilbert transform relative to weights. The main theorem appeals to the Ball and Helton almost-commutant lifting theorem to provide criteria for the existence of a solution to Nudel ′ ’ man’s problem.

We solve Gleason's problem in the reproducing kernel Hilbert space with repoducing kernel\(1/\left( {1 - \sum\nolimits_1^N {z_j w_j^* } } \right)\). We define and study some finite-dimensional resolvent-invariant subspaces that generalize the finite-dimensional de Branges-Rovnyak spaces to the setting of the ball.

We study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real Riemann surface; these spaces are analogues of the spaces introduced by L. de Branges when the Riemann sphere is replaced by a compact real Riemann surface of a higher genus. In de Branges theory an important role is played by resolvent-like difference quotient operators R α ; here we introduce generalized difference quotient operators R y α for any non-constant meromorphic function y on the Riemann surface. The spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators R y 1 α i and R y 2 α 2 , where y 1 and y 2 generate the field of meromorphic functions on the Riemann surface, which satisfy a supplementary identity, analogous to the de Branges identity for difference quotients. Just as the classical de Branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint (or nonunitary) operator, the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks.

Using reproducing kernel Hilbert spaces methods we develop a Schur-type algorithm for a subclass of the functions analytic and contractive in the ball. We also consider the Nevanlinna–Pick interpolation problem in that class.

We solve a boundary interpolation problem in the reproducing kernel Hilbert space of functions analytic in the unit ball of CN with reproducing kernel 1/(1−∑1Nzkwk*). We introduce the notion of Brune factor (or Blaschke–Potapov factor of the third kind) in this setting.

We prove a Beurling-Lax theorem for a family of reproducing kernel Hilbert spaces of functions analytic in an open subset of the unit ball containing the origin. The spaces under consideration are characterized by functions called Schur multipliers. Using the theory of linear relations in Pontryagin spaces we also give coisometric realizations of Schur multipliers. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 349-354.  2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS

We here study finite impulse response (FIR) rectangular, not necessarily causal, systems which are (para)-unitary on the unit circle (=the class $$\mathcal {U}$$ ). First, we offer three characterizations of these systems. Then, introduce a description of all FIRs in $$\mathcal {U}$$ , as copies of a real polytope, parametrized by the dimensions and the McMillan degree of the FIRs. Finally, we present six simple ways (along with their combinations) to construct, from any FIR, a large family of FIRs, of various dimensions and McMillan degrees, so that whenever the original system is in $$\mathcal {U}$$ , so is the whole family. A key role is played by Hankel matrices.

A boundary Nevanlinna–Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $${\kappa_1,\ldots,\kappa_N}$$ , quaternions p 1,…,p N all of modulus 1, so that the 2-spheres determined by each point do not intersect and p u ≠ 1 for u = 1,…, N, and quaternions s 1,…, s N , we wish to find a slice hyperholomorphic Schur function s so that $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}} s(r p_u) = s_u\quad \hbox{for}\ u=1,\ldots, N,$$ and $$\lim_{\substack{r\rightarrow 1\\ r \in(0,1)}}\frac{1-s(rp_u)\overline{s_u}}{1-r}\le\kappa_u,\quad\hbox{for}\ u=1,\ldots, N.$$ Our arguments rely on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.

In this paper we introduce and study some basic properties of the Fock space (also known as Segal–Bargmann space) in the slice hyperholomorphic setting. We discuss both the case of slice regular functions over quaternions and the case of slice monogenic functions with values in a Clifford algebra. In the specific setting of quaternions, we also introduce the full Fock space. This paper can be seen as the beginning of the study of infinitedimensional analysis in the quaternionic setting.

The aim of this paper is to show that we can extend the notion of convergence in the norm-resolvent sense to the case of several unbounded noncommuting operators (and to quaternionic operators as a particular case) using the notion of S-resolvent operator. With this notion, we can define bounded functions of unbounded operators using the S-functional calculus for n-tuples of noncommuting operators. The same notion can be extended to the case of the F-resolvent operator, which is the basis of the F-functional calculus, a monogenic functional calculus for n-tuples of commuting operators. We also prove some properties of the F-functional calculus, which are of independent interest. Copyright © 2013 John Wiley & Sons, Ltd.

The chapter deals with the definition of a bicomplex module and some structures induced on it by the properties of bicomplex scalars. It is shown how one can construct a bicomplex module beginning with a pair of complex linear spaces.

Using the notions and tools from realization in the sense of systems theory, we establish an explicit and new realization formula for families of infinite products of rational matrix-functions of a single complex variable. Our realizations of these resulting infinite products have the following four features: 1) Our infinite product realizations are functions defined in an infinite-dimensional complex domain. 2) Starting with a realization of a single rational matrix-function $M$, we show that a resulting infinite product realization obtained from $M$ takes the form of an (infinite-dimensional) Toeplitz operator with a symbol that is a reflection of the initial realization for $M$. 3) Starting with a subclass of rational matrix functions, including scalar-valued corresponding to low-pass wavelet filters, we obtain the corresponding infinite products that realize the Fourier transforms of generators of $\mathbf L_2(\mathbb R)$ wavelets. 4) We use both the realizations for $M$ and the corresponding infinite product to produce a matrix representation of the Ruelle-transfer operators used in wavelet theory. By matrix representation we refer to the slanted (and sparse) matrix which realizes the Ruelle-transfer operator under consideration., Comment: corrected version

The main properties of bicomplex and hyperbolic numbers are considered, in particular, the three conjugations on them generate the corresponding moduli of a bicomplex number which are not real valued: two of them are complex valued and one is hyperbolic valued. The notion of a positive hyperbolic number allows to introduce a partial order on the set of hyperbolic numbers which has far-reaching consequences in the theory of normed bicomplex modules.

We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space. Finally, we mention connections with the theory of discrete analytic functions, statistics, and with the quantum case.

This chapter treats, in the bicomplex context, the basic properties of linear functionals and linear operators. The notion of boundedness is given in terms of hyperbolic valued norms, thus allowing more similarity with the classical case. The bicomplex analogs of polarization identities are presented.

In this paper we prove a new version of Krein–Langer factorization theorem in the slice hyperholomorphic setting which is more general than the one proved in [9]. We treat both the case of functions with κ negative squares defined on subsets of the quaternionic unit ball or on subsets of the half-space of quaternions with positive real part. A crucial tool in the proof of our results is the Schauder–Tychonoff theorem and an invariant subspace theorem for contractions in a Pontryagin space.

We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle: (i) Through the realization matrix of Schur stable systems. (ii) The Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters. (iii) Through the (not necessarily reducible) Matrix Fraction Description (MFD). In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infinity). A special attention is devoted to exploring the gap between the square and rectangular cases., Comment: Improved version

Loewner's theorem on boundary interpolation of N κ functions is proved under rather general conditions. In particular, the hypothesis of Alpay and Rovnyak (1999) that the function f, which is to be extended to an N κ function, is defined and continuously differentiable on a nonempty open subset of the real line, is replaced by the hypothesis that the set on which f is defined contains an accumulation point at which f satisfies some kind of differentiability condition. The proof of the theorem in this note uses the representation of N κ functions in terms of selfadjoint relations in Pontryagin spaces and the extension theory of symmetric relations in Pontryagin spaces.

Resume Les fonctions de Caratheodory sont les fonctions analytiques et de partie positive reelle dans le demi-plan superieur C + . Elles sont associees a des espaces a noyau reproduisant (de noyau (2)) et interviennent en theorie des moments et dans la theorie de la prediction des processus stationnaires du second ordre. Nous etudions leurs analogues lorsque le disque unite est remplace par une surface de Riemann compacte reelle.

We use the theory of Hilbert spaces of analytic functions on bounded symmetric domains in C N to obtain information on the (1/(N+1)) st power of the Bergman kernel of the ball. This kernel has played recently an important and growing role in operator theory. We present several integral formulas for the Hilbert space generated by this kernel.

We study the reproducing kernel Hilbert spaces \(\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)\) with kernels of the form $$\frac{{I - S(z_1 ,z_2 > )S(w_1 ,w_2 )^* }}{{(1 - z_1 w_1^* )(1 - z_2 w_2^* )}}$$ where S(z1,z2) is a Schur function of two variables z1,z2l\(\mathbb{D}\). They are analogs of the spaces \(\mathfrak{H}\left( {\mathbb{D},S} \right)\) with reproducing kernel (1-S(z)S(w)*)/(1-zw*) introduced by de Branges and Rovnyak l. de Branges and J. Rovnyak, Square Summable Power Series Holt, Rinehart and Winston, New York, 1966. We discuss the characterization of \(\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)\) as a subspace of the Hardy space on the bidisk. The spaces \(\mathfrak{H}(\mathbb{D}^2 ,{\text{ }}S)\) form a proper subset of the class of the so–called sub–Hardy Hilbert spaces of the bidisk.

In the context of wide-sense stationary processes, the so-called Caratheodory–Fejer problem of extending a finite non-negative sequence of matrices has been much studied. We here investigate a similar extension problem in the setting of wide-sense periodically correlated processes: given the first N coefficients of T scalar-valued sequences, we study under which condition(s) it is possible to find T extensions which are the cyclocorrelaion sequences of a periodically correlated process with period T. Using a result of Gladysev, the problem is shifted to a Caratheodory–Fejer problem with symmetry constraints. The existence of extensions is proved. In nondegenerate cases, the set of all solutions is given in terms of a homographic transformation of some Schur function G. The choice G=0 leads to the maximum entropy solution. The associated Gaussian processes are then proved to have a periodic autoregressive structure.

We study a family of free stochastic processes whose covariance kernels K may be derived as a transform of a tempered measure σ . These processes arise, for example, in consideration of non-commutative analysis involving free probability. Hence our use of semi-circle distributions, as opposed to Gaussians. In this setting we find an orthonormal basis in the corresponding non-commutative L 2 of sample-space. We define a stochastic integral for our family of free processes.

Resume Le theoreme de Beurling caracterise les sous-espaces de l'espace de Hardy invariants par l'operateur de multiplication par z en termes de fonctions interieures. Dans cette Note nous considerons le cas ou la boule unite remplace le disque unite et l'espace a noyau reproduisant de noyau 1/(1− ∑ 1 N z j w j ∗ ) remplace l'espace de Hardy et donnons, dans le cas d'espaces de codimension finie, des formules explicites qui generalisent les produits de Blaschke.

Reproducing kernel spaces introduced by L. de Branges and J. Rovnyak provide isometric, coisometric and unitary realizations for Schur functions, i.e. for matrix-valued functions analytic and contractive in the open unit disk. In our previous paper [12] we showed that similar realizations exist in the “nonstationary setting”, i.e. when one considers upper triangular contractions (which appear in time-variant system theory as “transfer functions” of dissipative systems) rather than Schur functions and diagonal operators rather than complex numbers. We considered in [12] realizations centered at the origin. In the present paper we study realizations of a more general kind, centered at an arbitrary diagonal operator. Analogous realizations (centered at a point α of the open unit disk) for Schur functions were introduced and studied in [3] and [4].

We study the connections between the Caratheodory-Toeplitz extension problem and the Nehari extension problem in the discrete scalar case. This is the discrete counterpart of our previous paper [4].

It is well known that one may study Hardy spaces with the domain a finite bordered Riemann surface rather than the unit disk. However, for domains with more than one boundary component it is natural to consider, besides the usual positive definite inner product on H 2 , indefinite inner products obtained by picking up different signs (or in the vector-valued case, different signature matrices) when integrating over different components. In this paper we obtain a necessary and sufficient condition for such an indefinite inner product to be nondegenerate and show that when this condition is satisfied we actually get a Krein space. Furthermore, each holomorphic mapping of the finite bordered Riemann surface onto the unit disk (which maps boundary to boundary) determines an explicit isometric isomorphism between this space and a usual vector-valued Hardy space on the unit disk with an indefinite inner product defined by an appropriate hermitian matrix. As is usual when studying Hardy spaces on a multiply connected domain, the elements of the space are sections of a vector bundle rather than functions. The main point is to construct an appropriate extension of this bundle to the double of the finite bordered Riemann surface and to use Cauchy kernels for certain vector bundles on a compact Riemann surface.

In (D. Alpay, A. Chevreuil, Ph. Loubaton, J. Time Ser. Anal., 2000, to appear) an extension problem for covariance matrix of discrete-time periodically correlated stochastic processes introduced by Gladyshev was treated. In this paper we study the same problem for discrete-time almost periodically correlated stochastic processes. This problem can be reformulated and solved within the framework of interpolation for upper triangular operators. More precisely one can reduce the problem to an interpolation problem in the class of upper triangular operators of the Schur class.

Resume Les “sections de Brune” sont des fonctions matricielles rationnelles analytiques dans le disque unite D , J-interieures et qui ont un pole sur le cercle unite (voir la formule (1)). Elles ont ont un role important en theorie des systemes stationnaires. Dans cette note, nous presentons leur analogue dans le cadre des systemes non stationnaires, lorsque les fonctions analytiques et bornees dans D sont remplacees par des operateurs triangulaires superieurs bornes.

All solutions of one-sided tangential interpolation problems with Hilbert norm constraints for operator-valued Hardy functions on the polydisk are described. The minimal norm solution is explicitly expressed in terms of the interpolation data.

By a theorem of Loewner, a continuously differentiable real-valued function on a real interval whose difference quotient is a nonnegative kernel is the restriction of a holomorphic function which has nonnegative imaginary part in the upper half-plane and is holomorphic across the interval. An analogous result is obtained when the difference-quotient kernel has a finite number of negative squares. 1. CLASSICAL THEOREM OF LOEWNER In this paper we prove a version of Loewner's theorem for classes of meromorphic functions that generalize the Pick-Nevanlinna and Schur classes by a method that reduces the result to the classical case. In the present section the classical version of Loewner's theorem will be stated in two equivalent forms, for the unit circle and for the real line. Background results on reproducing kernel Pontryagin spaces and the generalized Schur and Nevanlinna classes So and No are discussed in ?2. The main results are stated and proved in ?3. The Pick class or Nevanlinna class is the set P of holomorphic functions which are defined and have nonnegative imaginary part on the open upper halfplane C+. The Schur class is the set S of holomorphic functions which are defined and bounded by one on the open unit disk D. The term kernel is understood to mean a Hermitian kernel, that is, a complex-valued function K(s, t) on a product set Q x Q such that K(t, s) = K(s, t) for all s, t E Q. We call a kernel nonnegative if all matrices (1.1) (K(sj, s)), sI, ..., s, E Q, n = 1, 2, 3, ..., are nonnegative. Let f (x) be a real-valued continuously differentiable function on a nonempty open subset G of the real line R. Define a kernel K(x, y) on G x G by (1.2) {x y) (z-z (A f'(x), X x=y. Received by the editors July 25, 1997. 1991 Mathematics Subject Classification. Primary 30E05, 47A57; Secondary 46C20, 47B50.

We continue our study of families of pairs of matrix-valued meromorphic functions P(ρ,P) depending on two parameters p and P introduced in [2]. These include as special cases the projective Schur, Nevanlinna and Caratheodory classes. A two sided Caratheodory Fejer interpolation problem is defined and solved in P(ρ,P), using the fundamental matrix inequality method. A corresponding Schur algorithm is studied. Finally we also consider the case of functions (as opposed to pairs).

The solutions of the Nevanlinna-Pick interpolation problem for generalized Stieltjes matrix functions are parametrized via a fractional linear transformation over a subset of the class of classical Stieltjes functions. The fractional linear transformation of some of these functions may have a pole in one or more of the interpolation points, hence not all Stieltjes functions can serve as a parameter. The set of excluded parameters is characterized in terms of the two related Pick matrices.

The Hilbert space of lower triangular Hilbert–Schmidt operators on the real line is a natural analogue of the Hardy space of a half-plane, where the complex numbers are now replaced by matrix-valued functions. One can associate with a bounded operator its “values” at a matrix-valued function [see Ballet al.,Oper. Theory Adv. Appl.56(1992), 52–89], and this allows [see Ballet al.,Integral Equations Operator Theory20(1994), 1–43] to define and solve the analogue of the two-sided Nudelman interpolation problem for bounded operators (which form an analogue ofH∞( C +)). In this paper we consider the two-sided interpolation problem with a Hilbert–schmidt norm constraint (rather than the more common operator-norm constraint) on the interpolant.

Motivated by the theory of nonstationary linear systems a number of problems in the theory of analytic functions have analogues in the setting of upper-triangular operators, where the complex variable is replaced by a diagonal operator. In this paper we focus on the analogue of interpolation in the Hardy space H2 and study a two-sided Nudelman type interpolation problem in the framework of upper-triangular Hilbert-Schmidt operators.

In this paper we give exact formulas for the potential associated to a Sturm-Liouville equation when the reflection coefficient function is rational. The solution is given in terms of a minimal realization of the reflection coefficient function.