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42
Spectral Factorization of Laurent Polynomials
 Advances in Computational Mathematics
, 1997
"... . We analyse the performance of five numerical methods for factoring a Laurent polynomial, which is positive on the unit circle, as the modulus squared of a real algebraic polynomial. It is found that there is a wide disparity between the methods, and all but one of the methods are significantly inf ..."
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Cited by 15 (1 self)
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. We analyse the performance of five numerical methods for factoring a Laurent polynomial, which is positive on the unit circle, as the modulus squared of a real algebraic polynomial. It is found that there is a wide disparity between the methods, and all but one of the methods are significantly influenced by the variation in magnitude of the coefficients of the Laurent polynomial, by the closeness of the zeros of this polynomial to the unit circle, and by the spacing of these zeros. Keywords: spectral factorization, Toeplitz matrices, EulerFrobenius polynomials, Daubechies wavelets. AMS subject classification: 12D05, 15A23. 1. Introduction Our recent interest in the asymptotic behaviour of the GramSchmidt iteration for orthonormalization of a large number of integer translates of a fixed function [11, 12] and also in techniques used for wavelet construction, cf. [7] and [20], has led us to experiment numerically with several existing algorithms to factor a Laurent polynomial, ass...
On The Solution Of WienerHopf Problems Involving Noncommutative Matrix Kernel Decompositions
 SIAM J. Appl. Math
, 1997
"... Many problems in physics and engineering with semiinfinite boundaries or interfaces are exactly solvable by the WienerHopf technique. It has been used successfully in a multitude of different disciplines when the WienerHopf functional equation contains a single scalar kernel. For complex boundary ..."
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Cited by 11 (7 self)
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Many problems in physics and engineering with semiinfinite boundaries or interfaces are exactly solvable by the WienerHopf technique. It has been used successfully in a multitude of different disciplines when the WienerHopf functional equation contains a single scalar kernel. For complex boundary value problems, however, the procedure often leads to coupled equations which therefore have a kernel of matrix form. The key step in the technique is to decompose the kernel into a product of two functions, one analytic in an upper region of a complex (transform) plane and the other analytic in an overlapping lower halfplane. This is straightforward for scalar kernels but no method has yet been proposed for general matrices. In this article a new procedure is introduced whereby Pade approximants are employed to obtain an approximate but explicit noncommutative factorization of a matrix kernel. As well as being simple to apply, the use of approximants allows the accuracy of the factorization to be increased almost indefinitely. The method is demonstrated by way of example. Scattering of acoustic waves by a semiinfinite screen at the interface between two compressible media with different physical properties is examined in detail. Numerical evaluation of the approximate factorizations together with results on an upper bound of the absolute error reveal that convergence with increasing Pade number is extremely rapid. The procedure is computationally efficient and of simple analytic form and offers high accuracy (error < 10 7 % typically). The method is applicable to a wide range of initial or boundary value problems.
Factorization of Almost Periodic Matrix Functions of Several Variables and Toeplitz Operators
 Math.Notes45 (1989), no. 5–6, 482–488. MR 90k:47033
"... this paper. We let (AP ..."
Spectral properties of random nonselfadjoint matrices and operators
, 2001
"... We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also desc ..."
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Cited by 7 (4 self)
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We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from selfadjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the nonselfadjoint Anderson model changes suddenly as one passes to the infinite volume limit.
Norms of Inverses, Spectra, and Pseudospectra of Large Truncated WienerHopf Operators and Toeplitz Matrices
, 1997
"... . This paper is concerned with WienerHopf integral operators on L p and with Toeplitz operators (or matrices) on l p . The symbols of the operators are assumed to be continuous matrix functions. It is well known that the invertibility of the operator itself and of its associated operator imply ..."
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Cited by 6 (2 self)
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. This paper is concerned with WienerHopf integral operators on L p and with Toeplitz operators (or matrices) on l p . The symbols of the operators are assumed to be continuous matrix functions. It is well known that the invertibility of the operator itself and of its associated operator imply the invertibility of all sufficiently large truncations and the uniform boundedness of the norms of their inverses. Quantitative statements, such as results on the limit of the norms of the inverses, can be proved in the case p = 2 by means of C algebra techniques. In this paper we replace C algebra methods by more direct arguments to determine the limit of the norms of the inverses and thus also of the pseudospectra of large truncations in the case of general p. Contents 1. Introduction 2 2. Structure of Inverses 4 3. Norms of Inverses 7 4. Spectra 15 5. Pseudospectra 17 6. Matrix Case 22 7. Block Toeplitz Matrices 23 8. Pseudospectra of Infinite Toeplitz Matrices 27 References 30 ...
LDU Factorization Results For BiInfinite And SemiInfinite Scalar And Block Toeplitz Matrices
, 1996
"... In this article various existence results for the LDUfactorization of semiinfinite and biinfinite scalar and block Toeplitz matrices and numerical methods for computing them are reviewed. Moreover, their application to the orthonormalization of splines is indicated. Both banded and nonbanded Toep ..."
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Cited by 6 (0 self)
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In this article various existence results for the LDUfactorization of semiinfinite and biinfinite scalar and block Toeplitz matrices and numerical methods for computing them are reviewed. Moreover, their application to the orthonormalization of splines is indicated. Both banded and nonbanded Toeplitz matrices are considered. Extensive use is made of matrix polynomial theory. Results on the approximation by the LDUfactorizations of finite sections are discussed. The generalization of the results to the LDUfactorization of multiindex Toeplitz matrices is outlined.
Holomorphic spaces: a brief and selective survey
 HOLOMORPHIC SPACES
, 1998
"... This article traces several prominent trends in the development of the subject of holomorphic spaces, with emphasis on operatortheoretic aspects. ..."
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Cited by 5 (0 self)
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This article traces several prominent trends in the development of the subject of holomorphic spaces, with emphasis on operatortheoretic aspects.
GENERALIZED KREIN ALGEBRAS AND ASYMPTOTICS OF TOEPLITZ DETERMINANTS
, 2006
"... This paper is dedicated to the centenary of Mark Krein (1907–1989). Abstract. We give a survey on generalized Krein algebras K α,β p,q and their applications to Toeplitz determinants. Our methods originated in a paper by Mark Krein of 1966, where he showed that K 1/2,1/2 2,2 is a Banach algebra. Sub ..."
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This paper is dedicated to the centenary of Mark Krein (1907–1989). Abstract. We give a survey on generalized Krein algebras K α,β p,q and their applications to Toeplitz determinants. Our methods originated in a paper by Mark Krein of 1966, where he showed that K 1/2,1/2 2,2 is a Banach algebra. Subsequently, Widom proved the strong Szegő limit theorem for block Toeplitz determinants with symbols in (K 1/2,1/2 2,2)N×N and later two of the authors studied symbols in the generalized Krein algebras (K α,β p,q)N×N, where λ: = 1/p+1/q = α+β and λ = 1. We here extend these results to 0 < λ < 1. The entire paper is based on fundamental work by Mark Krein, ranging from operator ideals through Toeplitz operators up to WienerHopf factorization. 1. Introduction and
BrunetDerrida particle systems, free boundary problems and WienerHopf equations
, 2009
"... We consider a branchingselection system in R with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N → ∞, the empirical measure process associated to the system converges in di ..."
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We consider a branchingselection system in R with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N → ∞, the empirical measure process associated to the system converges in distribution to a deterministic measurevalued process whose densities solve a free boundary integrodifferential equation. We also show that this equation has a unique traveling wave solution traveling at speed c or no such solution depending on whether c ≥ a or c < a, where a is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of WienerHopf equations. 1 Introduction and statement of the results We will consider the following branchingselection particle system. At any time t we have N particles on the real line with positions η N t (1) ≥ · · · ≥ η N t (N). Each one of the N particles gives birth at rate 1 to a new particle whose position is chosen, relative to the