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B: Pseudospectra of differential operators
 J. Oper. Theory
, 2000
"... We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely ..."
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Cited by 27 (7 self)
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We describe methods which have been used to analyze the spectrum of nonselfadjoint differential operators, emphasizing the differences from the selfadjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the operator. AMS subject classifications:34L05, 35P05, 47A10, 47A12
On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some SturmLiouville operators with square summable potential.
"... We prove that there exist some SturmLiouville operators with square summable potentials such that the singular continuous component of the spectral measure lies on the positive halfline. ..."
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Cited by 23 (2 self)
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We prove that there exist some SturmLiouville operators with square summable potentials such that the singular continuous component of the spectral measure lies on the positive halfline.
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 14 (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...
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 ..."
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 9 (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.
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 ...
8 J.STRUCKMEIER: Die Methode der finiten
 SIAM J. Appl. Math
, 1994
"... Abstract. We solve the twodimensional problem of acoustic scattering by a semiinfinite periodic array of identical isotropic point scatterers, i.e. objects whose size is negligible compared to the incident wavelength and which are assumed to scatter incident waves uniformly in all directions. This ..."
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Cited by 6 (5 self)
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Abstract. We solve the twodimensional problem of acoustic scattering by a semiinfinite periodic array of identical isotropic point scatterers, i.e. objects whose size is negligible compared to the incident wavelength and which are assumed to scatter incident waves uniformly in all directions. This model is appropriate for scatterers on which Dirichlet boundary conditions are applied in the limit as the ratio of wavelength to body size tends to infinity. The problem is also relevant to the scattering of an Epolarized electromagnetic wave by an array of highly conducting wires. The actual geometry of each scatterer is characterized by a single parameter in the equations, related to the the singlebody scattering problem and determined from a harmonic boundaryvalue problem. Using a mixture of analytical and numerical techniques, we confirm that a number of phenomena reported for specific geometries are in fact present in the general case (such as the presence of shadow boundaries in the far field and the vanishing of the circular wave scattered by the end of the array in certain specific directions). We show that the the semiinfinite array problem is equivalent to that of inverting an infinite Toeplitz matrix, which in turn can be formulated as a discrete WienerHopf problem. Numerical results are presented which compare the amplitude of the wave diffracted by the end of the array for scatterers having different shapes. Key words. Scattering; semiinfinite array; Foldy’s method; discrete WienerHopf. AMS subject classifications. 74J20,78A45. 1. Introduction. Many
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, in “Holomorphic spaces
 HANKEL OPERATORS. 19
, 1998
"... Abstract. This article traces several prominent trends in the development of the subject of holomorphic spaces, with emphasis on operatortheoretic aspects. The term “Holomorphic Spaces, ” the title of a program held at the Mathematical Sciences Research Institute in the fall semester of 1995, is sh ..."
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Cited by 5 (0 self)
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Abstract. This article traces several prominent trends in the development of the subject of holomorphic spaces, with emphasis on operatortheoretic aspects. The term “Holomorphic Spaces, ” the title of a program held at the Mathematical Sciences Research Institute in the fall semester of 1995, is short for “Spaces of Holomorphic Functions. ” It refers not so much to a branch of mathematics as to a common thread running through much of modern analysis—through functional analysis, operator theory, harmonic analysis, and, of course, complex analysis. This article will briefly outline the development of the subject from its origins in the early 1900’s to the present, with a bias toward operatortheoretic aspects, in keeping with the main emphasis of the MSRI program. I hope that the article will be accessible not only to workers in the field but to analysts in general. Origins The subject began with the thesis of P. Fatou [1906], a student of H. Lebesgue.