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Continuity of inner-outer factorization and cross sections from invariant subspaces to inner functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2024年第1期539卷

作者： Xin, Yue Hou, Bingzhe Jilin Univ Sch Math Changchun 130012 Peoples R China

Let H infinity be the Banach algebra of bounded analytic functions on the unit open disc D equipped with the supremum norm. As well known, inner functions play an important role in the study of bounded analytic functions. In this paper, we are interested in the study of inner functions. Following by the canonical inner-outer factorization decomposition, define Q inn and Q out the maps from H infinity to 0 the set of inner functions and F the set of outer functions, respectively. In this paper, we study the H 2-norm continuity and H infinity-norm discontinuity of Q inn and Q out on some subsets of H infinity . On the other hand, the Beurling theorem connects inner functions and invariant subspaces of the multiplication operator M z . We show the nonexistence of continuous cross section from some certain invariant subspaces to inner functions in the supremum norm. The continuity problem of Q inn and Q out on Hol(D), the set of all analytic functions in the closed unit disk, are considered. In addition, we also study the factor maps Q [ inn ] and Q [ out ] from H infinity to 0/T and F/ T, where 0/T is the quotient space of 0 mod T and F/T is the quotient space of F mod T. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： inner-outer factorization Bounded analytic functions Continuity Cross sections Invariant subspaces

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An inner-outer factorization in l^P with applications to ARMA processes

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2016年第1期437卷 396-418页

作者： Cheng, Raymond Ross, William T. Old Dominion Univ Dept Math & Stat Norfolk VA 23529 USA Univ Richmond Dept Math & Comp Sci Richmond VA 23173 USA

The following inner-outer type factorization is obtained for the sequence space F: if the complex sequence F = (F-0,F-1, F-2,...) decays geometrically, then for any l(p) sufficiently close to 2 there exist J and G in l(p) such that F = J * G;J is orthogonal in the Birkhoff James sense to all of its forward shifts SJ,S(2)J,S(3)J,...;J and F generate the same S -invariant subspace of F;and G is a cyclic vector for S on l(p). These ideas are used to show that an ARMA equation with characteristic roots inside and outside of the unit circle has Symmetric -a -Stable solutions, in which the process and the given white noise are expressed as causal moving averages of a related i.i.d. SaS white noise. An autoregressive representation of the process is similarly obtained. (C) 2016 Elsevier Inc. All rights reserved.

关键词： inner-outer factorization Hardy class Time series Autoregressive Moving average Stable process

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Identifying an autoregressive process disturbed by a moving-average noise using inner-outer factorization

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SIGNAL IMAGE AND VIDEO PROCESSING 2015年第1-Sup期9卷 235-244页

作者： Abdou, Ahmed Turcu, Flavius Grivel, Eric Diversi, Roberto Ferre, Guillaume Al Quds Univ Dept Elect & Commun Engn Jerusalem Israel Univ Bordeaux UMR CNRS 5218 Bordeaux INP ENSEIRB MATMECA IMS Cours Liberat 351 F-33405 Talence France Univ Bologna Dept Elect Elect & Informat Engn I-40136 Bologna Italy

This paper deals with the identification of an autoregressive (AR) process disturbed by an additive moving-average (MA) noise. Our approach operates as follows: Firstly, the AR parameters are estimated by using the overdetermined high-order Yule-Walker equations. The variance of the AR process driving process can be deduced by means of an orthogonal projection between two types of estimates of AR process correlation vectors. Then, the correlation sequence of the MA noise is estimated. Secondly, the MA parameters are obtained by using inner-outer factorization. To study the relevance of the resulting method, we compare it with existing algorithms, and we analyze the identifiability limits. The identification approach is then combined with Kalman filtering for channel estimation in mobile communication systems.

关键词： Autoregressive process (AR) Moving-avergae process (MA) inner-outer factorization Overdetermined high-order Yule-Walker equations

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factorization for finite subdiagonal algebras of type 1

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ARCHIV DER MATHEMATIK 2023年第2期120卷 183-194页

作者： Zhang, Ruihan Ji, Guoxing Shaanxi Normal Univ Sch Math & Stat Xian 710119 Peoples R China

Let A be a type 1 subdiagonal algebra in a finite von Neumann algebra M with respect to a faithful normal conditional expectation 4. We consider inner-outer factorization in noncommutative H-p (0 < p <= infinity) spaces associated with A. It is shown that for any nonzero x is an element of H-p, there exist a partial isometry V is an element of A and an outer h is an element of H-p such that x = Vh. Furthermore, we give a necessary and sufficient condition for a nonzero element in noncommutative L-p (M) to have a partial BN-factorization associated with A. As an application, we show that for any 0 < r, p, q <= infinity with 1/r = 1/p + 1/q, if h is an element of H-r, then there exist h(p) is an element of H-p and h(q) is an element of H-q such r p that h = h(p)h(q) and parallel to h parallel to(r) = parallel to h(p)parallel to(p) parallel to h(q)parallel to(q).

关键词： Von Neumarm algebra Type 1 subdiagonal algebra Noncommutative H-p space inner-outer factorization Partial BN-factorization

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FREE outer FUNCTIONS IN COMPLETE PICK SPACES

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 2023年第3期376卷 1929-1978页

作者： Aleman, Alexandru Hartz, Micheal Mccarthy, John E. Richter, Stefan Lund Univ Fac Sci Dept Math POB 118 S-22100 Lund Sweden Univ Saarland Fachrichtung Math D-66123 Saarbrucken Germany Washington Univ St Louis Dept Math One Brookings Dr St Louis MO 63130 USA Univ Tennessee Dept Math 1403 Circle DrKnoxville Tennessee TN 37996 USA

Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function f in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type f = phi g, where g is cyclic, phi is a contractive multiplier, and parallel to f parallel to = parallel to g parallel to. In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization.

关键词： Reproducing kernel Hilbert space inner-outer factorization

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An introduction to orthogonal filtering and its many applications 25

An introduction to orthogonal filtering and its many applica...

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25th IEEE Conference on Signal Processing - Algorithms, Architectures, Arrangements, and Applications (IEEE SPA)

作者： Dewilde, Patrick

ISBN: (数字)9788362065424

ISBN: (纸本)9788362065424

Orthogonal filtering is presented as a generic technique in signal processing, that is capable of solving many classical signal processing problems in a streamlined fashion, with emphasis on a time variant setting. Orthogonal filtering consists in efficient recursive orthogonalization of data, either original signal data or model data. Historically, it goes back to the notion of 'inner-outer factorization' in Hardy space theory, a notion that engineers refer to as 'decomposition of a transfer function in a lossless phase factor and a minimal phase, hence invertible, factor.' In contrast to the historical setting, the paper adopts a fully numerical, time variant approach. It starts out by developing the method on a 4 by 4 (block) example, which is then generalized to arbitrary discrete time, time variant systems. Next, the paper illustrates the utilization of the method in some detail on two classical problems: Kalman filtering and optimal quadratic control of a linear (time variant) system (a la Bellman). A number of further applications are mentioned and briefly discussed.

关键词： inner-outer factorization time variant systems Kalman filtering Optimal control

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Notch Filter Design based on Linear Quadratic Regulation Method 41

Notch Filter Design based on Linear Quadratic Regulation Met...

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第41届中国控制会议

作者： XingLou Zhu Cheng Wen School of Automation Science and Engineering South China University of Technology

ISBN: (数字)9789887581536

ISBN: (纸本)9781665482561

In this paper,a new design method of notch filter based on linear quadratic regulation(LQR) method is ***,according to the frequency domain explanations of the LQR cheap control and the observer based state feedback controller structure,the theoretical relationship between the optimal linear quadratic(LQ) controller and the notch filter is ***,the general steps of the proposed design method are detailed,which includes:system decompositions,optimal criterion design,and solution of the target notch *** addition,the feasibility of the proposed notch filter design method is verified by simulations,and the relationship between the notch performance and the parameter tuning is *** last,by compared with the traditional designed notch filter,the results show that the proposed optimal notch filter can reduce the resonance peak accurately and result in a better phase response of the compensated system.

关键词： notch filter cheap control inner-outer factorization servo control linear quadratic optimal

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Multiplication by finite Blaschke factors on a general class of Hardy spaces

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ADVANCES IN OPERATOR THEORY 2022年第4期7卷 1-26页

作者： Singh, Apoorva Sahni, Niteesh Shiv Nadar Inst Eminence Univ Dept Math Sch Nat Sci Gautam Buddh Nagar 201314 Uttar Pradesh India

A broader class of Hardy spaces and Lebesgue spaces have been introduced recently on the unit circle by considering continuous parallel *** to(1)-dominating normalized gauge norms instead of the classical norms on measurable functions. A Beurling type result has been proved for the operator of multiplication by the coordinate function. In this paper, we generalize the above Beurling type result to the context of multiplication by a finite Blaschke factor B(z) and also derive the common invariant subspaces of B-2(z) and B-3(z). These results lead to a factorization result for all functions in the Hardy space equipped with a continuous rotationally symmetric norm.

关键词： Normalized gauge norm Blaschke factor Invariant subspace Rotationally symmetric norm Hardy space Lebesgue space inner-outer factorization

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Stable state estimation in a network context ⁎

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IFAC-PapersOnLine 2024年第15期58卷 402-407页

作者： Robert R. Bitmead Department of Mechanical & Aerospace Engineering University of California San Diego La Jolla CA 92093-0411 USA

Power grid, communications, computer and product reticulation networks are frequently layered or subdivided by design. The OSI seven-layer computer network model and the electrical grid division into generation, transmission, distribution and associated markets are cases in point. The layering divides responsibilities and can be driven by operational, commercial, regulatory and privacy concerns. From a control context, a layer, or part of a layer, in a network isolates the authority to manage, i.e. control, a dynamic system with connections into unknown parts of the network. The topology of these connections is fully prescribed but the interconnecting signals are largely unavailable, through lack of sensing and even prohibition. Accordingly, one is driven to simultaneous input and state estimation methods. This is the province of this paper, guided by the structure of these network problems. We study a class of algorithms for this joint task, which if the system has transmission zeros outside the unit circle leads to an unstable and unimplementable estimator. Two modifications to the algorithm to ameliorate this problem were recently proposed involving (a) replacing the troublesome subsystem with its outer factor from its inner-outer factorization or (b) using the Kalman filter for a high-variance white signal model for the unknown inputs. The outer factor has only stable transmission zeros and so is stably invertible. The Kalman filter is stable by design. Here, we establish the connections between the original estimation problem for state and input signal and the outputs/estimates from the algorithm applied solely to the outer factor. We further show that both fixes coincide.

关键词： State estimation input estimation inner-outer factorization network estimation strong observability

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Norm of the evaluation on the vector-valued Hardy space

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2024年第1期531卷

作者： Kim, Sumin Sungkyunkwan Univ Dept Math Suwon 16419 South Korea

In this paper, we are concerned with the evaluation E-omega on the vector-valued Hardy space H-p(D, H) for a Hilbert space H. We show that for each omega is an element of D, the evaluation E-omega on H-p(D,H) is a bounded linear operator and||E-omega|| = (1/1 - |omega|(2))(1/p) for 1 <= p < infinity.(c) 2023 Elsevier Inc. All rights reserved.

关键词： Hardy spaces Evaluation inner-outer factorization

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