Inverses of Toeplitz Operators, Innovations, and Orthogonal Polynomials

Kailath, T.; Vieira, A.; Morf, M.

doi:10.1137/1020006

Cited by 291 publications

(76 citation statements)

References 24 publications

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“…and As, in general, \\P k \\ 2 # ||£*|| 2 for fe > /, the successive minima in (24) are no longer necessarily monotonie as k increases. Nevertheless, by considering the trial polynomial z k~J Pj{z)/tj in (24), we see by (4) that, for k > /, |CJ/|Cfc-il < h> whence (23) follows as earlier, although here Yim n _ J , o0 h n {G)/n need not exist.…”

Section: Itt J0mentioning

confidence: 98%

Maximum entropy and the moment problem

Landau¹

1987

Bull. Amer. Math. Soc.

120

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Introduction. The trigonometric moment problem stands at the source of several major streams in analysis. From it flow developments in function theory, in spectral representation of operators, in probability, in approximation, and in the study of inverse problems. Here we connect it also with a group of questions centering on entropy and prediction. In turn, this will suggest a simple approach, by way of orthogonal decomposition, to the moment problem itself.In statistical estimation, one often wants to guess an unknown probability distribution, given certain observations based on it. There are generally infinitely many distributions consistent with the data, and the question of which of these to select is an important one. The notion of entropy has been proposed here as the basis of a principle of salience which has received considerable attention. We will show that, in the context of spectral analysis, this idea is linked to a certain question of prediction by the trigonometric moment problem, and that all three strongly illuminate one another. The phenomena we describe are known, but our object is to unify them conceptually and to reduce the analytic intricacy of the arguments. To this end, we give a completely elementary discussion, virtually free of calculation, which shows that all the facts, including those concerning the moment problem, can be understood as direct consequences of orthogonal decomposition in a finite-dimensional space. We then describe how, in its continuous version, this leads to a view of second-order Sturm-Liouville differential equations, and conclude with some questions concerning the connection between combinatorial ideas and orthogonality in this problem.

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Section: Itt J0mentioning

confidence: 98%

Maximum entropy and the moment problem

Landau¹

1987

Bull. Amer. Math. Soc.

120

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“…Also, (JH) T (JH) = H T H. Thus, our results apply equally to Hankel and Toeplitz matrices. Our results might also be extended to more general classes of matrices with a displacement structure [50,51,52,53]. For simplicity we restrict our attention to the Toeplitz case.…”

Section: Introductionmentioning

confidence: 94%

Stability analysis of a general toeplitz systems solver

1995

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We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R T R is close to A T A. Thus, when the algorithm is used to solve the semi-normal equations R T Rx = A T b, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem min Ax − b 2 .

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“…and defining the reflectivity function 1d r(z) = 1 dz n z (z) (41) we obtain the two-component wave equations (1). The relations (40) …”

Section: Fii the Lossless Nonruniform Transmission Linementioning

confidence: 99%

“…Applying the operators x + and a to (83a) and (83b) respectively, and using the linearity of the resulting equations yields the Krein-Levinson recursions [15], [41] …”

Section: E[v(t)v(s)] = 6(t-s) (77b)mentioning

confidence: 99%

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The Schur algorithm and its applications

Yagle

Levy²

1985

Acta Appl Math

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The Schur algorithm and its times-domain counterpart, the fast Cholseky recursions, are some efficient signal processing algorithms which are well adapted to the study of inverse scattering problems.These algorithms use a layer stripping approach to reconstruct a lossless scattering medium described by symmetric two-component wave equations which model the interaction of right and left propagating waves.In this paper, the Schur and fast Cholesky recursions are presented and are used to study several inverse problems such as the reconstruction of nonuniform lossless transmission lines, the inverse problem for a layered acoustic medium, and the linear least-squares estimation of stationary stochastic processes. The inverse scattering problem for asymmetric two-component wave equations corresponding to lossy media is also examined and solved by using two coupled sets of Schur recursions. This procedure is then applied to the inverse problem for lossy transmission lines.-3-

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Inverses of Toeplitz Operators, Innovations, and Orthogonal Polynomials

Cited by 291 publications

References 24 publications

Maximum entropy and the moment problem

Maximum entropy and the moment problem

Stability analysis of a general toeplitz systems solver

The Schur algorithm and its applications

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