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The two-sided Lagrange–Sylvester interpolation problem introduced by Ball, Gohberg, and Rodman is solved in the framework ofH2functions, when some symmetry constraints are added.
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Letsbe a Schur function, that is a function analytic and contractive in the unit disk D. Then the function 1−s(z)s(ω)*/1−zω* is positive in D. L. de Branges and J. Rovnyak proved that the associated reproducing kernel Hilbert space... more
Letsbe a Schur function, that is a function analytic and contractive in the unit disk D. Then the function 1−s(z)s(ω)*/1−zω* is positive in D. L. de Branges and J. Rovnyak proved that the associated reproducing kernel Hilbert space provides the state space for a coisometric realization ofs. In a previous work we extended this result to the case of operator
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We prove a Gleason type theorem in the setting of functions hyperholomorphic in the unit ball of R4. We give an interpretation of the result in terms of pairs of functions defined in the unit ball of C2. Finally we use the theorem to... more
We prove a Gleason type theorem in the setting of functions hyperholomorphic in the unit ball of R4. We give an interpretation of the result in terms of pairs of functions defined in the unit ball of C2. Finally we use the theorem to study the homogeneous interpolation problem in the setting of hyperholomorphic functions. To cite this article: D. Alpay, M. Shapiro, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 889–894.
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We here specialize the standard matrix-valued polynomial interpolation to the case where on the imaginary axis the interpolating polynomials admit various symmetries: Positive semidefinite, Skew-Hermitian, $J$-Hermitian, Hamiltonian and... more
We here specialize the standard matrix-valued polynomial interpolation to the case where on the imaginary axis the interpolating polynomials admit various symmetries: Positive semidefinite, Skew-Hermitian, $J$-Hermitian, Hamiltonian and others. The procedure is comprized of three stages, illustrated through the case where on $i\R$ the interpolating polynomials are to be positive semidefinite. We first, on the expense of doubling the degree, obtain a minimal degree interpolating polynomial $P(s)$ which on $i\R$ is Hermitian. Then we find all polynomials $\Psi(s)$, vanishing at the interpolation points which are positive semidefinite on $i\R$. Finally, using the fact that the set of positive semidefinite matrices is a convex subcone of Hermitian matrices, one can compute the minimal scalar $\hat{\beta}\geq 0$ so that $P(s)+\beta\Psi(s)$ satisfies all interpolation constraints for all $\beta\geq\hat{\beta}$. This approach is then adapted to cases when the family of interpolating polynomials is not convex. Whenever convex, we parameterize all minimal degree interpolating polynomials.
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We consider bounded linear operators acting on the $\ell_2$ space indexed by the nodes of a homogeneous tree. Using the Cuntz relations between the primitive shifts on the tree, we generalize the notion of the single-scale time-varying... more
We consider bounded linear operators acting on the $\ell_2$ space indexed by the nodes of a homogeneous tree. Using the Cuntz relations between the primitive shifts on the tree, we generalize the notion of the single-scale time-varying point evaluation and introduce the corresponding reproducing kernel Hilbert space in which Cauchy's formula holds. These notions are then used in the study of the Schur multipliers and of the associated de Branges-Rovnyak spaces. As an application we obtain realization of Schur multipliers as transfer operators of multiscale input-state-output systems.
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We use the theory of reproducing kernel Hilbert spaces to solve a Carathéodory–Fejér interpolation problem in the class of Schur multipliers of the reproducing kernel Hilbert space of functions analytic in the unit ball of with... more
We use the theory of reproducing kernel Hilbert spaces to solve a Carathéodory–Fejér interpolation problem in the class of Schur multipliers of the reproducing kernel Hilbert space of functions analytic in the unit ball of with reproducing kernel .
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ABSTRACT We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-L\'evy type theorem and a factorization theorem. We give applications to... more
ABSTRACT We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-L\'evy type theorem and a factorization theorem. We give applications to Toeplitz and Wiener-Hopf operators.
ABSTRACT The analogue of the Riesz-Dunford functional calculus has been introduced and studied recently as well as the theory of semigroups and groups of linear quaternionic operators. In this paper we suppose that $T$ is the... more
ABSTRACT The analogue of the Riesz-Dunford functional calculus has been introduced and studied recently as well as the theory of semigroups and groups of linear quaternionic operators. In this paper we suppose that $T$ is the infinitesimal generator of a strongly continuous group of operators $(\mathcal{Z}_T(t))_{t \in \mathbb{R}}$ and we show how we can define bounded operators $f(T)$, where $f$ belongs to a class of functions which is larger than the class of slice regular functions, using the quaternionic Laplace-Stieltjes transform. This class will include functions that are slice regular on the $S$-spectrum of $T$ but not necessarily at infinity. Moreover, we establish the relation of $f(T)$ with the quaternionic functional calculus and we study the problem of finding the inverse of $f(T)$.
We prove representation theorems for Carath\'eodory functions in the setting of Banach spaces.
We study several aspects concerning slice regular functions mapping the quaternionic open unit ball into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive... more
We study several aspects concerning slice regular functions mapping the quaternionic open unit ball into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space. In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.
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We here extend the well known Positive Real Lemma (also known as the Kalman-Yakubovich-Popov Lemma) to complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. We then exploit this result... more
We here extend the well known Positive Real Lemma (also known as the Kalman-Yakubovich-Popov Lemma) to complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a by-product, we partition all state space realizations into
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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... more
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.
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... more
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 non-rational case, and establish new connections with the theory of generalized Schur functions and their associated reproducing kernel Pontryagin spaces, and the Cuntz relations.
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We here characterize the minimality of realization of arbitrary linear time-invariant dynamical systems through (i) intersection of the spectra of the realization matrix and of the corresponding state submatrix and (ii) moving the poles... more
We here characterize the minimality of realization of arbitrary linear time-invariant dynamical systems through (i) intersection of the spectra of the realization matrix and of the corresponding state submatrix and (ii) moving the poles by applying static output feedback. In passing, we introduce, for a given square matrix A, a parameterization of all matrices B for which the pairs (A, B) are controllable. In particular, the minimal rank of such B turns to be equal to the smallest geometric multiplicity among the eigenvalues of A. Finally, we show that the use of a (not necessarily square) realization matrix L to examine minimality of realization, is equivalent to the study of a smaller dimensions, square realization matrix L_sq, which in turn is linked to realization matrices obtained as polynomials in L_sq. Namely a whole family of systems.
ABSTRACT A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $\kappa_1, \ldots, \kappa_N$, quaternions \\ $p_1, \ldots, p_N$ all of modulus $1$, so that the... more
ABSTRACT A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $\kappa_1, \ldots, \kappa_N$, quaternions \\ $p_1, \ldots, p_N$ all of modulus $1$, so that the $2$-spheres determined by each point do not intersect and $p_u \neq 1$ for $u = 1,\ldots, N$, and quaternions $s_1, \ldots, s_N$, we wish to find a slice hyperholomorphic Schur function $s$ so that $$\lim_{r \uparrow 1} s(r p_u) = s_u\quad {\rm for} \quad u=1,\ldots, N,$$ and $$\lim_{r \uparrow 1} (1 - |s(r p_u)|^2)/(1-r^2) \leq \kappa_u \quad {\rm for} \quad u=1,\ldots, N.$$ We prove that a solution exists if and only if an analogue of the Pick matrix $P$ is positive semidefinite. In the case that $P$ is strictly positive definite all solutions are given by a linear fractional transformation. In the case that $P$ is singular the solution is unique. Our proof relies on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.
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ABSTRACT The aim of this note is to present a brief state of the art of Schur analysis in the slice hyperholomorphic setting.