Algorithms for fast multidimensional signal extraction

In recent studies operator factorization techniques have been developed for problems associated with multidimensional control and image processing. In particular the “special factorization” of Gohberg and Krein and the Schur-Coleski factorization have both been extended to the multivariate setting. The present study focuses on computational aspects of the earlier results. Also of interest are implications for designing computer architectures which facilitate high-speed, online computation of the operator factors.     

Outer factorizations in one and several variables

A multivariate version of Rosenblum's Fejér-Riesz theorem on outer factorization of trigonometric polynomials with operator coefficients is considered. Due to a simplification of the proof of the single variable case, new necessary and sufficient conditions for the multivariable outer factorization problem are formulated and proved.

     

FACTORIZATION OF UNBOUNDED THIN AND COTHIN OPERATORS

Abstract

Let X and Y be normed spaces and T: D(T) ? X → Y a linear operator. Following R.D. Neidingcr [N1] we recall the Davis, Figiel, Johnson, Pelczynski factorization of T corresponding to a parameter p (1 ≤ p ≤ ∞) and apply the corresponding factorization result in [N1] to unbounded thin operators. Properties equivalent to ubiquitous thinness arc derived. Defining an operator T to be cothin if its adjoint is thin, a dual factorization result for cothin operators is obtained, where for each 1 < p < ∞, the intermediate space in the factorization is cohereditarily l_p. This result is shown to hold more generally for the cases when T is either partially continuous or closable; in particular, such operators are strictly cosingular. A condition for a closable weakly compact operator to be strictly cosingular follows as a corollary.     

On the far-field operator in elastic obstacle scattering

We investigate the far-field operator for the scattering oftime-harmonic elastic plane waves by either a rigid body, acavity, or an absorbing obstacle. Extending results of Colton& Kress for acoustic obstacle scattering, for the spectrumof the far-field operator we show that there exist an infinitenumber of eigenvalues and determine disks in the complex planewhere these eigenvalues lie. In addition, as counterpart ofan identity in acoustic scattering due to Kress & Päivärinta,we will establish a factorization for the difference of thefar-field operators for two different scatterers. Finally, extendinga sampling method for the approximate solution of the acousticinverse obstacle scattering problem suggested by Kirsch to elasticity,this factorization is used for a characterization of a rigidscatterer in terms of the eigenvalues and eigenelements of thefar-field operator.     

Multigrid for second‐order ADN elliptic systems

This paper theoretically examines a multigrid strategy for solving systems of elliptic partial differential equations (PDEs) introduced in the work of Lee. Unlike most multigrid solvers that are constructed directly from the whole system operator, this strategy builds the solver using a factorization of the system operator. This factorization is composed of an algebraic coupling term and a diagonal (decoupled) differential operator. Exploiting the factorization, this approach can produce decoupled systems on the coarse levels. The corresponding coarse‐grid operators are in fact the Galerkin variational coarsening of the diagonal differential operator. Thus, rather than performing delicate coarse‐grid selection and interpolation weight procedures on the original strongly coupled system as often done, these procedures are isolated to the diagonal differential operator. To establish the theoretical results, however, we assume that these systems of PDEs are elliptic in the Agmon–Douglis–Nirenberg (ADN) sense and apply the factorization and multigrid only to the principal part of the system of PDEs. Two‐grid error bounds are established for the iteration applied to the complete system of PDEs. Numerical results are presented to illustrate the effectiveness of this strategy and to expose factors that affect the convergence of the methods derived from this strategy.     

Meromorphic Factorization Revisited and Application to Some Groups of Matrix Functions

Some properties and applications of meromorphic factorization of matrix functions are studied. It is shown that a meromorphic factorization of a matrix function G allows one to characterize the kernel of the Toeplitz operator with symbol G without actually having to previously obtain a Wiener–Hopf factorization. A method to turn a meromorphic factorization into a Wiener–Hopf one which avoids having to factorize a rational matrix that appears, in general, when each meromorphic factor is treated separately, is also presented. The results are applied to some classes of matrix functions for which the existence of a canonical factorization is studied and the factors of a Wiener–Hopf factorization are explicitly determined. Submitted: April 15, 2007. Revised: October 26, 2007. Accepted: December 12, 2007.     

On Factorization of Trigonometric Polynomials

We give a new proof of the operator version of the Fejér-Riesz Theorem using only ideas from elementary operator theory. As an outcome, an algorithm for computing the outer polynomials that appear in the Fejér-Riesz factorization is obtained. The extremal case, where the outer factorization is also *-outer, is examined in greater detail. The connection to Aglers model theory for families of operators is considered, and a set of families lying between the numerical radius contractions and ordinary contractions is introduced. The methods are also applied to the factorization of multivariate operator-valued trigonometric polynomials, where it is shown that the factorable polynomials are dense, and in particular, strictly positive polynomials are factorable. These results are used to give results about factorization of operator valued polynomials over , in terms of rational functions with fixed denominators.     

Maharam-type kernel representation for operators with a trigonometric domination

Consider a linear and continuous operator T between Banach function spaces. We prove that under certain requirements an integral inequality for T is equivalent to a factorization of T through a specific kernel operator: in other words, the operator T has what we call a Maharam-type kernel representation. In the case that the inequality provides a domination involving trigonometric functions, a special factorization through the Fourier operator is given. We apply this result to study the problem that motivates the paper: the approximation of functions in \(L^{2}[0,1]\) by means of trigonometric series whose Fourier coefficients are given by weighted trigonometric integrals.     

Constructively Factoring Linear Partial Differential Operators in Two Variables

We study conditions under which a partial differential operator of arbitrary order n in two variables or an ordinary linear differential operator admits a factorization with a first-order factor on the left.The process of factoring consists of recursively solving systems of linear equations subject to certain differential compatibility conditions.In the general case of partial differential operators, it is not necessary to solve a differential equation. In special degenerate cases, such as an ordinary differential operator, the problem eventually reduces to solving some Riccati equation(s). We give the factorization conditions explicitly for the second and third orders and in outline form for higher orders. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 165–180, November, 2005.     

Quadratic optimal control of stable well-posed linear systems

We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.

     

Asymptotic behavior of solutions to nonlinear Volterra integral equations

A class of operator Riccati integral equations is associated with a factorization problem in a certain Banach algebra. Recent results concerning factorization in this algebra are used to obtain existence, uniqueness, and continuous dependence results for the Riccati equations.     

On the generalized invertibility of Wiener-Hopf operators in Banach spaces

It is proved that a Wiener-Hopf operator T_p (A) on a Banach space PX is generalized invertible iff A has a cross factorization with respect toX and P. IfX is a separable Hilbert space, then a criterion for the weak factorization of A can be concluded.     

变系数线性微分方程的算子解法

首先讨论变系数线性微分算子因式分解式的存在条件 ,并且给出微分算子因式分解的一些技巧 ,然后给出变系数线性微分方程算子解法的两种方法 .     

On the triangular factorization of positive operators

An operator integral, referred to as the amplitude integral (AI) and used in the BC-method (based on boundarycontrol theory) for solving inverse problems, is systematically studied. For a continuous operator and two families of increasing subspaces, the continual analog of the matrix diagonal in the form of an AI is introduced. The convergence of the AI is discussed. An example of an operator with no diagonal is provided. The role of the diagonal in the problem of triangular factorization is elucidated. The well-known result of matrix theory stating the uniqueness of triangular factorization with a prescribed diagonal is extended. It is shown that the corresponding factor can be represented in the AI form. The correspondence between the AI and the classical representation of the triangular factor of an operator that is a sum of the identity and a compact operator is established.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 45–60.This work was supported by INTAS under grant INTAS 93-1815 and by the Russian Foundation for Basic Research under grant 96-01-00666. The authors are grateful to Prof. S. N. Naboko for fruitful discussions and consultations.     

Integration of the Continual System of Nonlinear Interaction Waves

We derive the continual system of nonlinear interaction waves from the compatibility of the transport equation on the whole line and the equation governing the time-evolution of the eigenfunctions of the transport operator. The transport equation represents the continual generalization from the n-component system of first-order ordinary differential equations. The continual system describes a nonlinear interaction of waves. We prove that the continual system can be integrated by the inverse scattering method. The method is based on applying the results of the inverse scattering problem for the transport equation to finding the solution of the Cauchy initial-value problem for the continual system. Indeed, the transition operator for the scattering problem admits right and left Volterra factorizations. The intermediate operator for this problem determines the one-to-one correspondence between the preimages of a solution of the transport equation. This operator is related to the transition operator and admits not only right and left Volterra factorizations but also the analytic factorization. By virtue of this fact the potential in the transport equation is uniquely reconstructed in terms of the solutions of the fundamental equations in inverse problem.We introduce the generalized Lax equation. This enables us to derive the time-evolution of the transition operator. Then, the time-dependent intermediate operator is constructed. The solution of the considered Cauchy problem is expressed in terms of solutions of the fundamental equations in inverse problem. This solution is found uniquely from the given initial condition.     

On operator order and chaotic operator order for two operators

In this paper we characterize operator order A?B?O and chaotic operator order log A?logB for positive and invertible operators A and B in terms of operator inequalities via the Furuta inequality and operator equalities due to the Douglas’s majorization and factorization. Related results are obtained, which include generalizations and characterizations of some well-known results.     

The factorization method for the Askey–Wilson polynomials

A special Infeld–Hull factorization is given for the Askey–Wilson second order q-difference operator. It is then shown how to deduce a generalization of the corresponding Askey–Wilson polynomials.     

On some new boundary-value problems in nonsmooth domains

The wave factorization of an elliptic pseudodifferential operator and the solvability of the equation with such an operator are described.     

Rank factorizations of positive operators

The notion of a rank factorization of a positive operator through a cone is introduced and related to the nonnegative rank factorization of a nonnegative matrix. The concept is applied to the study of group inverses of certain positive operators.