An Operator Perspective on Signals and Systems

Frazho, A. E.; Bhosri, Wisuwat

doi:10.1007/978-3-0346-0292-1

Cited by 24 publications

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“…We shall derive the above lemma as a corollary of the following somewhat more general lemma. The additional invertibility condition appearing in the final sentences of the above two lemmas is always fulfilled if U is finite dimensional; see Remark 3.2.3 in [21]. Moreover, this invertibility condition is also satisfied if Φ = Υ 22 , where Υ 22 is given by (1.19).…”

Section: This Proves (A32)mentioning

confidence: 92%

All Solutions to an Operator Nevanlinna–Pick Interpolation Problem

Frazho

Horst

Kaashoek³

2018

Operator Theory in Different Settings and Related Applications

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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 variation 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.2010 Mathematics Subject Classification. Primary 47A57; Secondary 47A48, 47A56, 47A62, 28D20.for more details. Then for F ∈ S(U, Y), the operator W T F is also a controllability operator, namely for the pair {Z,Although the LTONP interpolation problem has a simple formulation, it covers two relevant special cases that will be discussed in Sections 8 and 9 below. In both cases it is essential that we do not demand that the spectral radius is strictly less than one. In Section 8 we discuss a large class of commutant lifting problems that can be written in the form of a LTONP interpolation problem. Conversely, any LTONP interpolation problem can be rewritten as a commutant lifting problem from this specific class. Hence the problems are equivalent in this sense. In this case, the operator Z will be a compression of a unilateral forward shift operator and will typically not have spectral radius less than one. The connection with commutant lifting is already observed in [19, Section II.2] and also appears in the more general setting of the Drury-Arveson space in [8].The second special case, discussed in Section 9, is the Leech problem. This problem, and its solution, originates from a paper by R.B. Leech, which was written in 1971-1972, but published only recently [40]; see [35] for an account of the history behind this paper. The Leech problem is another nontrivial example of a LTONP interpolation problem for which the operator Z need not have spectral radius less than one, in fact, in this case, the operator Z is equal to a unilateral forward shift operator and hence its spectral radius is equal to one. Our analysis of the rational Leech problem [22,23,24] inspired us to study in detail the class of LTONP interpolation problems. It led to new results and improvements on our earlier results on the Leech problem.Next we will present our main results. This requires some preparation. Let {W, W , Z} be a LTONP data set. Set P = W W * and P = W W * . The intertwining relations in (1.1) imply thatHe...

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Section: This Proves (A32)mentioning

confidence: 92%

All Solutions to an Operator Nevanlinna–Pick Interpolation Problem

Frazho

Horst

Kaashoek³

2018

Operator Theory in Different Settings and Related Applications

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“…Also note that δ(G) = rank H G < ∞ and δ(K) = rank H K < ∞ imply that the subspace N Φ of Lemma 2.1 is finite dimensional, and hence the subspace M Φ in (2.2) is finite dimensional, since ker T * Φ = {0}. Then Theorem 4.3.2 in [8] yields that the inner function Θ associated with M Φ is a two-sided inner rational matrix function, that is, Θ ∈ RH ∞ r×r and ΘΘ * = Θ * Θ is identically equal to I r . The next proposition provides the relations between the McMillan degrees given in Theorem 0.2.…”

Section: Sanne Ter Horstmentioning

confidence: 96%

“…The first two identities in (3.5) say that Φ is a right and F a left spectral factors of R. The last identity, together with Θ two-sided inner, provides a Douglas-ShapiroShields factorization of Φ, cf., [8, Chapter 4].…”

Section: Moreover In This Casementioning

confidence: 99%

“…Hence Φ = ΘF * . It is well known that T R > 0 holds if and only if its outer spectral factor is invertible outer, c.f., [8 …”

Section: Here δ(G) and δ(K) Denote The Mcmillan Degrees Of G And K Rmentioning

confidence: 99%

“…Note that there are effective ways to computing Kolmogorov decompositions, cf., [6]. Moreover, the functions R, Φ, Θ and F can be computed explicitly using state space techniques from mathematical systems theory (cf., [4,8]), starting from a state space representation of the function [ G K ]. This will be the topic of a forthcoming paper of the present author together with A.E.…”

Section: Introductionmentioning

confidence: 99%

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Rational matrix solutions to the Leech equation: The Ball–Trent approach revisited

Horst

2013

Journal of Mathematical Analysis and Applications

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Abstract. Using spectral factorization techniques, a method is given by which rational matrix solutions to the Leech equation with rational matrix data can be computed explicitly. This method is based on an approach by J.A. Ball and T.T. Trent, and generalizes techniques from recent work of T.T. Trent for the case of polynomial matrix data. IntroductionConsider H ∞ -matrix functions G ∈ H K is positive. Note that (0.1) is equivalent to T G T X = T K and T X ≤ 1. Hence Leech's theorem can be viewed as the analogue of the Douglas factorization lemma [7] within the class of analytic Toeplitz operators. The necessity of (0.2) follows directly from Douglas' factorization lemma and the reformulation of (0.1) in terms of Toeplitz operators. The other implication is more involved. The solution criterion (0.2) can also be formulated directly in terms of the functions G and K, it is equivalent to the mapbeing a positive kernel in the sense of Aronszajn [1], that is, for any finite sequence z 1 , . . . , z n ∈ D the block operator matrix [L(z i , z j )] i,j=1,...,n defines a positive operator on the Hilbert space direct sum of n copies of C m . We note that the actual result by Leech is stated in the general context of Hilbert space operators intertwining shift operators, and in particular holds for operator-valued H ∞ -functions as well. Our interest is primarily in the case where G and K are rational matrix functions.There exists various proofs of Leech's theorem, see [10] and the references therein. In [3] Ball and Trent prove a generalization of Leech's theorem to the polydisc in C d , adapting a technique coined the 'lurking isometry' approach in [2], and give aKey words and phrases. Leech equation, Toeplitz operators, stable rational matrix functions, outer spectral factor. 1 2 SANNE TER HORST description of all X ∈ H ∞ p×q satisfying (0.1). We briefly outline the construction here, specified to the single variable case.The positivity of T G T *. Such a factorization is often referred to as a Kolmogorov decomposition in the literature, cf., [6]. LetΛ • be the analytic operator-valued function on D, with valuesΛ From this identity one derives the existence of a partial isometryThis in turn implies that the function X defined on D byis in H From the point of view of rational matrix functions the above construction has one disadvantage. In general, the Hilbert space H • appearing in (0.5) is infinite dimensional, and in that case it is hard to see when the solution X in (0.7) is rational. In fact, even if both G and K are rational matrix functions,may very well be of infinite rank. More precisely, see Theorem 3.2 below, in the rational matrix case* for all 0 ≤ t ≤ 2π. Overcoming this difficulty is the main theme of the present paper.In the context of the Toeplitz-corona problem, which can be reduced to the special case of (0.1) with q = m and K(z) = I m , z ∈ D, Trent [12] deduced a modification of the above procedure for the special case that G is a row vector (m = 1) polynomial, leading to a rational column vector sol...

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Infinite‐time partially observed nonzero‐sum bilinear affine‐quadratic stochastic differential game and its application to competitive advertising

Tao

Zhang

Yang

2023

Asian Journal of Control

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Due to the conjunction of huge expenditure and latent inefficiencies, competitive advertising has always been occupying a front‐and‐center place in marketing, among which a bilinear system is one of the most important. Because much real business is two giants playing among other emerging challengers and the players can only inspect the market through a related noisy process by the random sampling survey, the sales‐advertising dynamics is a partially observed nonzero‐sum stochastic differential game. However, by the intrinsic difficulties of mathematics, there is no extant research to augment the analysis on the infinite‐time horizon, which is of particular significance to implementing the online sales campaign and evaluating the consequence of endless competition. Motivated by this, controllability and stability analysis are explored first for the iterative approximation to bilinearity. Then, the simulation provides a rapid prototyping and automatic approach for the most efficient strategies. Moreover, four implications have been discovered by comparison with eight examples as the application of four encounters on finite‐ and infinite‐time horizons. Several new insights have been revealed for those struggling to develop successful advertising strategies.

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An Operator Perspective on Signals and Systems

Cited by 24 publications

References 0 publications

All Solutions to an Operator Nevanlinna–Pick Interpolation Problem

All Solutions to an Operator Nevanlinna–Pick Interpolation Problem

Rational matrix solutions to the Leech equation: The Ball–Trent approach revisited

Infinite‐time partially observed nonzero‐sum bilinear affine‐quadratic stochastic differential game and its application to competitive advertising

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