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Two-Dimensional Moment Problem and schur algorithm

INTEGRAL EQUATIONS AND OPERATOR THEORY 2025年第1期97卷 1-21页

作者： Kovalyov, Ivan Kunis, Stefan Univ Osnabruck Albrechtstr 28A D-49076 Osnabruck Germany

We study a truncated two-dimensional moment problem in terms of the Stieltjes transform. The set of the solutions is described by the schur step-by-step algorithm, which is based on the continued fraction expansion of the solution. In particular, the obtained results are applicable to the two-dimensional moment problem for atomic measures.

关键词： Continued fractions Multidimensional moment problem schur algorithm Atomic measure

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schur algorithm in The Class SI of J-contractive Functions Intertwining Solutions of Linear Differential Equations

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INTEGRAL EQUATIONS AND OPERATOR THEORY 2012年第3期74卷 313-344页

作者： Alpay, Daniel Melnikov, Andrey Vinnikov, Victor Ben Gurion Univ Negev Dept Math IL-84105 Beer Sheva Israel Drexel Univ Philadelphia PA 19104 USA

In the PhD thesis of the second author under the supervision of the third author was defined the class of SI of J-contractive functions, depending on a parameter and arising as transfer functions of overdetermined conservative 2D systems invariant in one direction. In this paper we extend and solve in the class SI, a number of problems originally set for the class S of functions contractive in the open right-half plane, and unitary on the imaginary line with respect to some preassigned signature matrix J. The problems we consider include the schur algorithm and the Nevanlinna-Pick interpolation problem. The arguments rely on a correspondence between elements in a given subclass of SI and elements in S. Another important tool in the arguments is a new result pertaining to the classical tangential schur algorithm.

关键词： schur algorithm Time-invariant 2D-systems Linear differential equations

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The schur algorithm Applied to the One-Dimensional Continuous Inverse Scattering Problem

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2013年第13期61卷 3311-3320页

作者： Choi, Youngchol Chun, Joohwan Kim, Taejoon Bae, Jinho Korea Inst Ocean Sci & Technol Taejon 305343 South Korea Korea Adv Inst Sci & Technol Dept Elect Engn Taejon 305701 South Korea City Univ Hong Kong Dept Elect Engn Kowloon Hong Kong Peoples R China Jeju Natl Univ Dept Ocean Syst Engn Cheju 690756 South Korea

The one-dimensional continuous inverse scattering problem can be solved by the schur algorithm in the discrete-time domain using sampled scattering data. The sampling rate of the scattering data should be increased to reduce the discretization error, but the complexity of the schur algorithm is proportional to the square of the sampling rate. To improve this tradeoff between the complexity and the accuracy, we propose a schur algorithm with the Richardson extrapolation (SARE). The asymptotic expansion of the schur algorithm, necessary for the Richardson extrapolation, is derived in powers of the discretization step, which shows that the accuracy order (with respect to the discretization step) of the schur algorithm is 1. The accuracy order of the SARE with the N-step Richardson extrapolation is increased to N + 1 with comparable complexity to the schur algorithm. Therefore, the discretization error of the schur algorithm can be decreased in a computationally efficient manner by the SARE.

关键词： Inverse scattering reflection coefficient Richardson extrapolation schur algorithm

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Circulant matrix factorization based on schur algorithm for designing optical multimirror filters

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JAPANESE JOURNAL OF APPLIED PHYSICS PART 1-REGULAR PAPERS BRIEF COMMUNICATIONS & REVIEW PAPERS 2006年第6A期45卷 5163-5168页

作者： Bae, Jinho Jeju Natl Univ Coll Ocean Sci Cheju 690756 Jeju South Korea

We propose a schur-type algorithm that includes spectral factorization of covariance matrix using circulant matrix factorization to design optical multimirror filters. The schur algorithm is the method used for a fast Cholesky factorization of the Toeplitz matrix, which can determine the reflection coefficient of optical multimirror structures. Circulant matrix factorization is a very powerful tool used for spectral factorization from the covariance polynomial using matrix manipulation in vector space that can be found in the minimum phase polynomials without using the polynomial root finding method. We present a detailed description of the circulant matrix factorization for the reciprocal polynomial approximation of an arbitrary curve (or spectrum). The schur algorithm can, in turn, be applied to obtain the desired reflection coefficient of the optical filters. We also verify the performance of the proposed method by comparing it with the polynomial root finding method.

关键词： circulant matrix factorization schur algorithm optical multimirror Toeplitz

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The schur algorithm and reproducing kernel Hilbert spaces in the ball

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LINEAR ALGEBRA AND ITS APPLICATIONS 2002年第1-3期342卷 163-186页

作者： Alpay, D Bolotnikov, V Kaptanoglu, HT Ben Gurion Univ Negev Dept Math IL-84105 Beer Sheva Israel Coll William & Mary Dept Math Williamsburg VA 23187 USA Middle E Tech Univ Dept Math TR-06531 Ankara Turkey

Using reproducing kernel Hilbert spaces methods we develop a schur-type algorithm for a subclass of the functions analytic and contractive in the ball. We also consider the Nevanlinna-Pick interpolation problem in tha... 详细信息

关键词： schur algorithm Leech's theorem unit ball

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CHEBYSHEV-POLYNOMIAL-BASED schur-algorithm

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IEE PROCEEDINGS-F RADAR AND SIGNAL PROCESSING 1990年第1期137卷 11-14页

作者： CHAPMAN, R RAHMAN, MAA UNIV AUSIT DEPT ELECT & ELECTR ENGNAUSITEGYPT

The paper presents a new version of the well known schur algorithm. The schur algorithm may be used in a wide range of signal-processing applications, from stability tests for discrete time polynomials, through inverse problems and speech coding to the design of orthogonal digital filters. Since the algorithm is iterative in nature there is a tendency for roundoff errors to accumulate through the iterations to the point where the schur algorithm can become unpractical in certain applications. The design of orthogonal lattice filters is an example of this. The paper expands the polynomials used in the schur algorithm in terms of Chebyshev polynomials, and reformulates the schur algorithm in this Chebyshev domain. It is shown that this can lead to smaller roundoff errors than the classical algorithm.

关键词： orthogonal digital filters iterative speech coding polynomials Algebra digital filters inverse problems orthogonal lattice filters Chebyshev polynomials Interpolation and function approximation (numerical analysis) schur algorithm roundoff errors Digital filters signal-processing filtering and prediction theory signal processing Chebyshev approximation discrete time polynomials stability Information theory Signal processing and detection

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The schur algorithm applied to the design of optical multi-mirror structures

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2005年第2-3期12卷 283-292页

作者： Bae, J Chun, J Kailath, T Korea Adv Inst Sci & Technol Dept Elect Engn Taejon 305701 South Korea Stanford Univ Informat Syst Lab Stanford CA 94305 USA

We show how to use the classical schur algorithm to design multi-mirror optical interferometers (or filters). Our derivation is simple and straightforward, clearly revealing its connection to the previously known orthogonal digital filter structures. We also give a complete detailed description of an FFT-based algorithm for the reciprocal polynomial approximation of an arbitrary curve (or spectrum). The schur algorithm can, in turn, be applied to the obtained polynomial to get the desired reflection coefficients of the mirrors. Copyright (c) 2004 John Wiley & Sons, Ltd.

关键词： schur algorithm optical filter orthogonal filter FFT

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The schur algorithm for an indefinite Stieltjes moment problem

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MATHEMATISCHE NACHRICHTEN 2017年第11-12期290卷 1637-1662页

作者： Derkach, Volodymyr Kovalyov, Ivan Dragomanov Natl Pedag Univ Dept Math Pirogova 9 UA-01601 Kiev Ukraine Donetsk Natl Univ Dept Math 600 Richchya Str 21 UA-21021 Vinnytsya Ukraine

The nondegenerate truncated indefinite Stieltjes moment problem in the class N-k(k) of generalized Stieltjes functions is considered. To describe the set of solutions of this problem we apply the schur step-by-step algorithm, which leads to the expansion of these solutions in generalized Stieltjes continuous fractions studied recently in [11]. Explicit formula for the resolvent matrix in terms of generalized Stieltjes polynomials is found. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

关键词： Indefinite Stieltjes moment problem generalized Stieltjes function continued fraction generalized Stieltjes fraction schur algorithm solution matrix

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Tensor algebras and displacement structure - I. The schur algorithm

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ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN 2002年第1期21卷 3-20页

作者： Constantinescu, T Johnson, JL Univ Texas Richardson TX 75083 USA

In this paper we explore the connection between tensor algebras and displacement structure. Thus, we describe a scattering experiment in this framework, we obtain a realization of the elements of the tenser algebra as transfer maps of a certain class of nonstationary linear systems, and we describe a schur type algortihm for the schur elements of the tensor algebra.

关键词： displacement structure tensor algebras schur algorithm

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A schur algorithm for computing matrix pth roots

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2003年第4期24卷 971-989页

作者： Smith, MI Univ Manchester Dept Math Manchester M13 9PL Lancs England

Any nonsingular matrix has pth roots. One way to compute matrix pth roots is via a specialized version of Newton's method, but this iteration has poor convergence and stability properties in general. A schur algorithm for computing a matrix pth root that generalizes methods of Bjorck and Hammarling [Linear Algebra Appl., 52/53 (1983), pp. 127-140] and Higham [Linear Algebra Appl., 88/89 (1987), pp. 405-430] for the square root is presented. The algorithm forms a schur decomposition of A and computes a pth root of the (quasi-) triangular factor by a recursion. The backward error associated with the schur method is examined, and the method is shown to have excellent numerical stability.

关键词： matrix pth root schur algorithm Newton's method commutativity stability real arithmetic rounding error analysis

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