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On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some SturmLiouville operators with square summable potential
 J. Differential Equations
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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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. 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...
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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. 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 ...
Spectral analysis of subordinate Brownian motions in halfline. Preliminary version
 Preprint, 2010, arXiv:1006.0524v1. SUBORDINATE BROWNIAN MOTIONS IN HALFLINE 57
"... Abstract. We study onedimensional Lévy processes with LévyKhintchine exponent ψ(ξ2), where ψ is a complete Bernstein function. These processes are subordinate Brownian motions corresponding to subordinators, whose Lévy measure has completely monotone density; or, equivalently, symmetric Lévy proce ..."
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Abstract. We study onedimensional Lévy processes with LévyKhintchine exponent ψ(ξ2), where ψ is a complete Bernstein function. These processes are subordinate Brownian motions corresponding to subordinators, whose Lévy measure has completely monotone density; or, equivalently, symmetric Lévy processes whose Lévy measure has completely monotone density on (0,∞). Examples include symmetric stable processes and relativistic processes. The main result is a formula for the generalized eigenfunctions of transition operators of the process killed after exiting the halfline. A generalized eigenfunction expansion of the transition operators is derived. As an application, a formula for the distribution of the first passage time (or the supremum functional) is obtained.
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 14 (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
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Quantitative theory of ordinary differential equations and tangential Hilbert 16th problem
 DEPARTMENT OF MATHEMATICS, WEIZMANN INSTITUTE OF SCIENCE, P.O.B. 26, REHOVOT 76100, ISRAEL EMAIL ADDRESS: YAKOV@WISDOM.WEIZMANN.AC.IL WWW
, 2001
"... These highly informal lecture notes aim at introducing and explaining several closely related problems on zeros of analytic functions defined by ordinary differential equations and systems of such equations. The main incentive for this study was its potential application to the tangential Hilbert 1 ..."
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Cited by 12 (6 self)
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These highly informal lecture notes aim at introducing and explaining several closely related problems on zeros of analytic functions defined by ordinary differential equations and systems of such equations. The main incentive for this study was its potential application to the tangential Hilbert 16th problem on zeros of complete Abelian integrals. The exposition consists mostly of examples illustrating various phenomena related to this problem. Sometimes these examples give an insight concerning the proofs, though the complete exposition of the latter is mostly relegated to separate expositions.
Maximum recoverable work, minimum free energy and state space in linear viscoelasticity
 Quarterly of Applied Mathematics
"... Abstract. The various formulations of the maximum recoverable work used in literature are proved to be equivalent. Then an explicit formula of the minimum free energy is derived starting from the formulation of the maximum recoverable work given by Day. The resulting expression is equivalen to that ..."
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Abstract. The various formulations of the maximum recoverable work used in literature are proved to be equivalent. Then an explicit formula of the minimum free energy is derived starting from the formulation of the maximum recoverable work given by Day. The resulting expression is equivalen to that found by Golden and other authors. However the particular formulation allows to prove that the domain of definition of minimum free energy is the whole state space. Finally the maximum recoverable work is shown to be put as the basis of the thermodynamics of viscoelastic materials under isothermal condidtions. In this context the usual relation between the ClausiusDuhem inequality and the dissipation of the material is restored. 1. Introduction. All the definitions of Helmholtz free energy for a viscoelastic material (for instance the one given by Graffi [11, 18, 19, 20] and that stated by Coleman and Owen [3, 7, 8]) do not identify a unique functional. Moreover it has been shown [23] that in the convex set of all free energies there exist maximum and minimum element.
NonEuclidean functional analysis and electronics
 Bull. Amer. Math. Soc. (N.S
, 1982
"... —functional analysis, a few symmetric spaces, a Lie group over a function field, and NevanlinnaPick interpolation theory—all fixed on a framework of engineering motivation which determines the relationship between them. A large branch of functional analysis concerns linear spaces of analytic functi ..."
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Cited by 11 (0 self)
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—functional analysis, a few symmetric spaces, a Lie group over a function field, and NevanlinnaPick interpolation theory—all fixed on a framework of engineering motivation which determines the relationship between them. A large branch of functional analysis concerns linear spaces of analytic functions
An explicit formula for the minimum free energy in linear viscoelasricity
 J. of Elasticity
"... Abstract. A general explicit formula for the maximum recoverable work from a given state is derived in the frequency domain for full tensorial isothermal linear viscoelastic constitutive equations. A variational approach, developed for the scalar case, is here generalized by virtue of certain factor ..."
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Abstract. A general explicit formula for the maximum recoverable work from a given state is derived in the frequency domain for full tensorial isothermal linear viscoelastic constitutive equations. A variational approach, developed for the scalar case, is here generalized by virtue of certain factorizability properties of positivedefinite matrices. The resultant formula suggests how to characterize the state in the sense of Noll in the frequency domain. The property that the maximum recoverable work represents the minimum free energy according to both Graffi’s and ColemanOwen’s definitions is used to obtain an explicit formula for the minimum free energy. Detailed expressions are presented for particular types of relaxation function.