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Joseph A. Ball - One of the best experts on this subject based on the ideXlab platform.
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de branges rovnyak spaces and norm constrained Interpolation
arXiv: Classical Analysis and ODEs, 2014Co-Authors: Joseph A. Ball, Vladimir BolotnikovAbstract:For $S$ a contractive analytic operator-valued function on the unit disk ${\mathbb D}$, de Branges and Rovnyak associate a Hilbert space of analytic functions ${\mathcal H}(S)$. A companion survey provides equivalent definitions and basic properties of these spaces as well as applications to function theory and operator theory. The present survey brings to the fore more recent applications to a variety of more elaborate function theory Problems, including $H^\infty$-norm constrained Interpolation, connections with the Potapov method of Fundamental Matrix Inequalities, parametrization for the set of all solutions of an Interpolation Problem, variants of the Abstract Interpolation Problem of Katsnelson, Kheifets, and Yuditskii, boundary behavior and boundary Interpolation in de Branges-Rovnyak spaces themselves, and extensions to multivariable and Kre\u{\i}n-space settings.
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Abstract Interpolation in vector-valued de Branges-Rovnyak spaces
Integral Equations and Operator Theory, 2010Co-Authors: Joseph A. Ball, Vladimir Bolotnikov, Sanne Ter HorstAbstract:Following ideas from the Abstract Interpolation Problem of Katsnelson et al. (Operators in spaces of functions and Problems in function theory, vol 146, pp 83-69, Naukova Dumka, Keiv, 1987) for Schur class functions, we study a general metric constrained Interpolation Problem for functions from a vector-valued de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with an operator-valued Schur class function $S$. A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm. Attention is also paid to the case that the map which provides this description is injective. The Interpolation Problem studied here contains as particular cases (1) the vector-valued version of the Interpolation Problem with operator argument considered recently in Ball et al. (Proc Am Math Soc 139(2), 609-618, 2011) (for the nondegenerate and scalar-valued case) and (2) a boundary Interpolation Problem in $\mathcal{H}(K_S)$. In addition, we discuss connections with results on kernels of Toeplitz operators and nearly invariant subspaces of the backward shift operator.
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a constrained nevanlinna pick Interpolation Problem for matrix valued functions
Indiana University Mathematics Journal, 2010Co-Authors: Joseph A. Ball, Vladimir Bolotnikov, Sanne Ter HorstAbstract:Recent results of Davidson-Paulsen-Raghupathi-Singh give neces- sary and sufficient conditions for the existence of a solutionto the Nevanlinna- Pick Interpolation Problem on the unit disk with the additional restriction that the interpolant should have the value of its derivative at the origin equal to zero. This concrete mild generalization of the classical Problem is prototypi- cal of a number of other generalized Nevanlinna-Pick Interpolation Problems which have appeared in the literature (for example, on a finitely-connected pla- nar domain or on the polydisk). We extend the results of Davidson-Paulsen- Raghupathi-Singh to the setting where the interpolant is allowed to be matrix- valued and elaborate further on the analogy with the theory of Nevanlinna-Pick Interpolation on a finitely-connected planar domain.
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Interpolation Problems for schur multipliers on the drury arveson space from nevanlinna pick to abstract Interpolation Problem
Integral Equations and Operator Theory, 2008Co-Authors: Joseph A. Ball, Vladimir BolotnikovAbstract:We survey various increasingly more general operator-theoretic formulations of generalized left-tangential Nevanlinna-Pick Interpolation for Schur multipliers on the Drury-Arveson space. An adaptation of the methods of Potapov and Dym leads to a chain-matrix linear-fractional parametrization for the set of all solutions for all but the last of the formulations for the case where the Pick operator is invertible. The last formulation is a multivariable analogue of the Abstract Interpolation Problem formulated by Katsnelson, Kheifets and Yuditskii for the single-variable case; we obtain a Redheffer-type linear-fractional parametrization for the set of all solutions (including in degenerate cases) via an adaptation of ideas of Arov and Grossman.
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the abstract Interpolation Problem and commutant lifting a coordinate free approach
2000Co-Authors: Joseph A. Ball, Tavan T TrentAbstract:We present a coordinate-free formulation of the Abstract Interpolation Problem introduced by Katsnelson, Kheifets and Yuditskii in an abstract scattering theory framework. We also show how the commutant lifting theorem fits into this new formulation of the Abstract Interpolation Problem, giving a coordinate-free version of a result of Kupin.
A. E. Frazho - One of the best experts on this subject based on the ideXlab platform.
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all solutions to an operator nevanlinna pick Interpolation Problem
Operator Theory: Advances and Applications, 2018Co-Authors: A. E. Frazho, S Ter Horst, M A KaashoekAbstract:The main results presented in this paper provide a complete and explicit description of all solutions to the left tangential operator Nevanlinna– Pick Interpolation Problem assuming the associated Pick operator is strictly positive. The complexity of the solutions is similar to that found in descriptions of the sub–optimal Nehari Problem and variations on the Nevanlinna– Pick Interpolation Problem in the Wiener class that have been obtained through the band method. The main techniques used to derive the formulas are based on the theory of co-isometric realizations, and use the Douglas factorization lemma and state space calculations. A new feature is that we do not assume an additional stability assumption on our data, which allows us to view the Leech Problem and a large class of commutant lifting Problems as special cases. Although the paper has partly the character of a survey article, all results are proved in detail and some background material has been added to make the paper accessible to a large audience including engineers.
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multirate filterbank design a relaxed commutant lifting approach
IEEE Transactions on Signal Processing, 2010Co-Authors: Wisuwat Bhosri, A. E. FrazhoAbstract:In this paper, we reformulate the design of the IIR synthesis filters in classical multirate systems as an Interpolation Problem involving a norm called the P m norm where m is any positive integer. This Interpolation Problem can be solved using relaxed commutant lifting techniques in operator theory. The P m norm is actually a tradeoff in handling energy distortion and error peak distortion. Our development allows the designer to select from a family of filters the one which is best suited for a specific application. The well-known H 2 and H ? design methods can be viewed as special cases when m = 1 and m ? ? respectively. The computation relies mainly on FFT techniques and a finite section of certain Toeplitz matrices. The resulting filters are given in state space form and maybe useful for practical implementation.
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a time variant norm constrained Interpolation Problem arising from relaxed commutant lifting
Operator algebras operator theory and applications, 2009Co-Authors: A. E. Frazho, S Ter Horst, M A KaashoekAbstract:A time-variant analogue of an Interpolation Problem equivalent to the relaxed commutant lifting Problem is introduced and studied. In a somewhat less general form the Problem already appears in the analysis of the set of all solutions to the three chain completion Problem. The interpolants are upper triangular operator matrices of which the columns induce contractive operators. The set of all solutions of the Problem is described explicitly. The results presented are time-variant analogues of the main theorems in [23].
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a time variant norm constrained Interpolation Problem arising from relaxed commutant lifting
arXiv: Functional Analysis, 2008Co-Authors: A. E. Frazho, S Ter Horst, M A KaashoekAbstract:A time-variant analogue of an Interpolation Problem equivalent to the relaxed commutant lifting Problem is introduced and studied. In a somewhat less general form the Problem already appears in the analysis of the set of all solutions to the three chain completion Problem. The interpolants are upper triangular operator matrices of which the columns induce contractive operators. The set of all solutions of the Problem is described explicitly. The results presented are time-variant analogues of the main theorems in [A.E. Frazho, S. ter Horst, and M.A. Kaashoek, All solutions to the relaxed commutant lifting Problem, Acta Sci. Math. (Szeged) 72 (2006), 299--318].
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relaxed commutant lifting an equivalent version and a new application
arXiv: Functional Analysis, 2007Co-Authors: A. E. Frazho, S Ter Horst, M A KaashoekAbstract:This paper presents a few additions to commutant lifting theory. An operator Interpolation Problem is introduced and shown to be equivalent to the relaxed commutant lifting Problem. Using this connection a description of all solutions of the former Problem is given. Also a new application, involving bounded operators induced by $H^2$ operator-valued functions, is presented.
Vladimir Bolotnikov - One of the best experts on this subject based on the ideXlab platform.
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de branges rovnyak spaces and norm constrained Interpolation
arXiv: Classical Analysis and ODEs, 2014Co-Authors: Joseph A. Ball, Vladimir BolotnikovAbstract:For $S$ a contractive analytic operator-valued function on the unit disk ${\mathbb D}$, de Branges and Rovnyak associate a Hilbert space of analytic functions ${\mathcal H}(S)$. A companion survey provides equivalent definitions and basic properties of these spaces as well as applications to function theory and operator theory. The present survey brings to the fore more recent applications to a variety of more elaborate function theory Problems, including $H^\infty$-norm constrained Interpolation, connections with the Potapov method of Fundamental Matrix Inequalities, parametrization for the set of all solutions of an Interpolation Problem, variants of the Abstract Interpolation Problem of Katsnelson, Kheifets, and Yuditskii, boundary behavior and boundary Interpolation in de Branges-Rovnyak spaces themselves, and extensions to multivariable and Kre\u{\i}n-space settings.
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self mappings of the quaternionic unit ball multiplier properties schwarz pick inequality and nevanlinna pick Interpolation Problem
arXiv: Complex Variables, 2013Co-Authors: Daniel Alpay, Vladimir Bolotnikov, Fabrizio Colombo, Irene SabadiniAbstract: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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Abstract Interpolation in vector-valued de Branges-Rovnyak spaces
Integral Equations and Operator Theory, 2010Co-Authors: Joseph A. Ball, Vladimir Bolotnikov, Sanne Ter HorstAbstract:Following ideas from the Abstract Interpolation Problem of Katsnelson et al. (Operators in spaces of functions and Problems in function theory, vol 146, pp 83-69, Naukova Dumka, Keiv, 1987) for Schur class functions, we study a general metric constrained Interpolation Problem for functions from a vector-valued de Branges-Rovnyak space $\mathcal{H}(K_S)$ associated with an operator-valued Schur class function $S$. A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm. Attention is also paid to the case that the map which provides this description is injective. The Interpolation Problem studied here contains as particular cases (1) the vector-valued version of the Interpolation Problem with operator argument considered recently in Ball et al. (Proc Am Math Soc 139(2), 609-618, 2011) (for the nondegenerate and scalar-valued case) and (2) a boundary Interpolation Problem in $\mathcal{H}(K_S)$. In addition, we discuss connections with results on kernels of Toeplitz operators and nearly invariant subspaces of the backward shift operator.
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a constrained nevanlinna pick Interpolation Problem for matrix valued functions
Indiana University Mathematics Journal, 2010Co-Authors: Joseph A. Ball, Vladimir Bolotnikov, Sanne Ter HorstAbstract:Recent results of Davidson-Paulsen-Raghupathi-Singh give neces- sary and sufficient conditions for the existence of a solutionto the Nevanlinna- Pick Interpolation Problem on the unit disk with the additional restriction that the interpolant should have the value of its derivative at the origin equal to zero. This concrete mild generalization of the classical Problem is prototypi- cal of a number of other generalized Nevanlinna-Pick Interpolation Problems which have appeared in the literature (for example, on a finitely-connected pla- nar domain or on the polydisk). We extend the results of Davidson-Paulsen- Raghupathi-Singh to the setting where the interpolant is allowed to be matrix- valued and elaborate further on the analogy with the theory of Nevanlinna-Pick Interpolation on a finitely-connected planar domain.
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Interpolation Problems for schur multipliers on the drury arveson space from nevanlinna pick to abstract Interpolation Problem
Integral Equations and Operator Theory, 2008Co-Authors: Joseph A. Ball, Vladimir BolotnikovAbstract:We survey various increasingly more general operator-theoretic formulations of generalized left-tangential Nevanlinna-Pick Interpolation for Schur multipliers on the Drury-Arveson space. An adaptation of the methods of Potapov and Dym leads to a chain-matrix linear-fractional parametrization for the set of all solutions for all but the last of the formulations for the case where the Pick operator is invertible. The last formulation is a multivariable analogue of the Abstract Interpolation Problem formulated by Katsnelson, Kheifets and Yuditskii for the single-variable case; we obtain a Redheffer-type linear-fractional parametrization for the set of all solutions (including in degenerate cases) via an adaptation of ideas of Arov and Grossman.
M A Kaashoek - One of the best experts on this subject based on the ideXlab platform.
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all solutions to an operator nevanlinna pick Interpolation Problem
Operator Theory: Advances and Applications, 2018Co-Authors: A. E. Frazho, S Ter Horst, M A KaashoekAbstract:The main results presented in this paper provide a complete and explicit description of all solutions to the left tangential operator Nevanlinna– Pick Interpolation Problem assuming the associated Pick operator is strictly positive. The complexity of the solutions is similar to that found in descriptions of the sub–optimal Nehari Problem and variations on the Nevanlinna– Pick Interpolation Problem in the Wiener class that have been obtained through the band method. The main techniques used to derive the formulas are based on the theory of co-isometric realizations, and use the Douglas factorization lemma and state space calculations. A new feature is that we do not assume an additional stability assumption on our data, which allows us to view the Leech Problem and a large class of commutant lifting Problems as special cases. Although the paper has partly the character of a survey article, all results are proved in detail and some background material has been added to make the paper accessible to a large audience including engineers.
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a time variant norm constrained Interpolation Problem arising from relaxed commutant lifting
Operator algebras operator theory and applications, 2009Co-Authors: A. E. Frazho, S Ter Horst, M A KaashoekAbstract:A time-variant analogue of an Interpolation Problem equivalent to the relaxed commutant lifting Problem is introduced and studied. In a somewhat less general form the Problem already appears in the analysis of the set of all solutions to the three chain completion Problem. The interpolants are upper triangular operator matrices of which the columns induce contractive operators. The set of all solutions of the Problem is described explicitly. The results presented are time-variant analogues of the main theorems in [23].
-
a time variant norm constrained Interpolation Problem arising from relaxed commutant lifting
arXiv: Functional Analysis, 2008Co-Authors: A. E. Frazho, S Ter Horst, M A KaashoekAbstract:A time-variant analogue of an Interpolation Problem equivalent to the relaxed commutant lifting Problem is introduced and studied. In a somewhat less general form the Problem already appears in the analysis of the set of all solutions to the three chain completion Problem. The interpolants are upper triangular operator matrices of which the columns induce contractive operators. The set of all solutions of the Problem is described explicitly. The results presented are time-variant analogues of the main theorems in [A.E. Frazho, S. ter Horst, and M.A. Kaashoek, All solutions to the relaxed commutant lifting Problem, Acta Sci. Math. (Szeged) 72 (2006), 299--318].
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relaxed commutant lifting an equivalent version and a new application
arXiv: Functional Analysis, 2007Co-Authors: A. E. Frazho, S Ter Horst, M A KaashoekAbstract:This paper presents a few additions to commutant lifting theory. An operator Interpolation Problem is introduced and shown to be equivalent to the relaxed commutant lifting Problem. Using this connection a description of all solutions of the former Problem is given. Also a new application, involving bounded operators induced by $H^2$ operator-valued functions, is presented.
Jean B Lasserre - One of the best experts on this subject based on the ideXlab platform.
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lmis for constrained polynomial Interpolation with application in trajectory planning
International Conference on Robotics and Automation, 2004Co-Authors: Didier Henrion, Jean B LasserreAbstract:We consider an open-loop trajectory planning Problem for linear systems with bound constraints originating from saturations or physical limitations. Using an algebraic approach and results on positive polynomials, we show that this control Problem can be cast into a constrained polynomial Interpolation Problem admitting a convex linear matrix inequality (LMI) formulation
