Infinite Product Representations for Kernels and Iterations of Functions

Alpay, Daniel; Jørgensen, Palle E. T.; Lewkowicz, Izchak; Itzik, Martziano,

doi:10.1007/978-3-319-10335-8_5

Cited by 12 publications

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References 16 publications

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“…Although composition of rational functions plays an important role in the theory of dynamical systems (see e.g. [5], [12]), a few associated questions are yet unsolved. We here touch upon three aspects.…”

Section: Introductionmentioning

confidence: 99%

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Composition of rational functions: State-space realization and applications

Alpay

Lewkowicz

2019

Linear Algebra and its Applications

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We define two versions of compositions of matrix-valued rational functions of appropriate sizes and whenever analytic at infinity, offer a set of formulas for the corresponding state-space realization, in terms of the realizations of the original functions. Focusing on positive real functions, the first composition is applied to electrical circuits theory along with introducing a connection to networks of feedback loops. The second composition is applied to Stieltjes functions.AMS Classification: 08A02 26C15 37F10 47B33 47N70 94C05

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Section: Introductionmentioning

confidence: 99%

“…Out = (F + G −1 ) −1 · In Now, one can identify F and G inFigure 4, with Y F and Z G respectively, fromFigure 3.As an engineering application of item (III) of Proposition 3.6 seeFigures 2,5,6. …”

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confidence: 99%

Composition of rational functions: State-space realization and applications

Alpay

Lewkowicz

2019

Linear Algebra and its Applications

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“…Following the work in [1], if these operators satisfy the Cuntz relations, S * i S j = δ ij I and S i S * i = I, then the functions given by S i1 · · · S im 1(z), where m ∈ N, i j ∈ {0, 1}, and 1(z) = 1, form an orthonormal basis for the reproducing kernel Hilbert space associated to the kernel function K(z, w). It was shown in [7] that this is the case for the family of polynomials F = {az 2 n+2 − 2az 2 n+1 : a = 0}, if we take the exponent α in K(z, w) to be 2 n .…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Orthonormal Bases Associated to Basins of Attraction

Tipton

2017

Complex Anal. Oper. Theory

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Abstract. In [1], a technique was developed which allows for the construction of a reproducing kernel Hilbert space on basins of attraction containing 0. When the right conditions are met, an explicit orthonormal basis can be constructed using a particular class of operators. It is natural then to consider how the orthonormal basis changes as we let the basin of attraction vary. We will consider this question for the basins of attraction containing 0 of the family of polynomials F = {az 2 n+2 − 2az 2 n+1 : a = 0}, where n ∈ N.

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“…We refer the reader to the papers [4][5][6][7][8], and also see [9,10]. Of more recent papers dealing with results which have motivated our present paper are [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

“…We shall now return to a stochastic variation of formula (20), the so called Malliavin derivative in the direction k. In this, the system (a * (h 1 ) , · · · , a * (h n )) in Equation (20) instead takes the form of a multivariate Gaussian random variable.…”

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confidence: 99%

Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus

Jørgensen

Tian

2016

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Abstract:We study densely defined unbounded operators acting between different Hilbert spaces. For these, we introduce a notion of symmetric (closable) pairs of operators. The purpose of our paper is to give applications to selected themes at the cross road of operator commutation relations and stochastic calculus. We study a family of representations of the canonical commutation relations (CCR)-algebra (an infinite number of degrees of freedom), which we call admissible. The family of admissible representations includes the Fock-vacuum representation. We show that, to every admissible representation, there is an associated Gaussian stochastic calculus, and we point out that the case of the Fock-vacuum CCR-representation in a natural way yields the operators of Malliavin calculus. We thus get the operators of Malliavin's calculus of variation from a more algebraic approach than is common. We further obtain explicit and natural formulas, and rules, for the operators of stochastic calculus. Our approach makes use of a notion of symmetric (closable) pairs of operators. The Fock-vacuum representation yields a maximal symmetric pair. This duality viewpoint has the further advantage that issues with unbounded operators and dense domains can be resolved much easier than what is possible with alternative tools. With the use of CCR representation theory, we also obtain, as a byproduct, a number of new results in multi-variable operator theory which we feel are of independent interest.

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Infinite Product Representations for Kernels and Iterations of Functions

Cited by 12 publications

References 16 publications

Composition of rational functions: State-space realization and applications

Composition of rational functions: State-space realization and applications

Dynamics of Orthonormal Bases Associated to Basins of Attraction

Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus

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