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# Toeplitz matrix

About: Toeplitz matrix is a(n) research topic. Over the lifetime, 8097 publication(s) have been published within this topic receiving 131901 citation(s).

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Abstract: Part I: Toeplitz Forms: Preliminaries Orthogonal polynomials. Algebraic properties Orthogonal polynomials. Limit properties The trigonometric moment problem Eigenvalues of Toeplitz forms Generalizations and analogs of Toeplitz forms Further generalizations Certain matrices and integral equations of the Toeplitz type Part II: Applications of Toeplitz Forms: Applications to analytic functions Applications to probability theory Applications to statistics Appendix: Notes and references Bibliography Index.

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2,279 citations

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01 Jan 1977-

TL;DR: The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toepler matrices with absolutely summable elements are derived in a tutorial manner in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject.

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Abstract: The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.

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2,231 citations

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01 Jan 1984-

Abstract: Part I: Toeplitz Forms: Preliminaries Orthogonal polynomials. Algebraic properties Orthogonal polynomials. Limit properties The trigonometric moment problem Eigenvalues of Toeplitz forms Generalizations and analogs of Toeplitz forms Further generalizations Certain matrices and integral equations of the Toeplitz type Part II: Applications of Toeplitz Forms: Applications to analytic functions Applications to probability theory Applications to statistics Appendix: Notes and references Bibliography Index.

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1,561 citations

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TL;DR: This work considers least-squares problems where the coefficient matrices A,b are unknown but bounded and minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A.

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Abstract: We consider least-squares problems where the coefficient matrices A,b are unknown but bounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A,b are rational functions of an unknown-but-bounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation.

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1,083 citations

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01 Jan 1990-

Abstract: Preface to Volume II Table of contents of Volume II Introduction PART V: TRIANGULAR REPRESENTATIONS XX Additive lower-upper triangular decompositions of operators 1 Additive lower-upper triangular decompositions relative to finite chains 2 Preliminaries about chains 3 Diagonals 4 Chains on Hilbert space 5 Triangular algebras 6 Riemann-Stieltjes integration along chains 7 Additive lower-upper decomposition theorem 8 Additive lower-upper decomposition of a Hilbert-Schmidt operator 9 Multiplicative integration along chains 10 Basic properties of reproducing kernel Hilbert spaces and chains 11 Example of an additive LU-decomposition in a RKHS XXI Operators in triangular form 1 Triangular representation 2 Intermezzo about completely nonselfadjoint operators 3 Volterra operators with a one-dimensional imaginary part 4 Unicellular operators XXII Multiplicative lower-upper triangular decompositions of operators 1 LU-factorization with respect to a finite chain 2 The LU-factorization theorem 3 LU-factorizations of compact perturbations of the identity 4 LU-factorizatioris of Hilbert-Schmidt perturbations of the identity 5 LU-factorizations of integral operators 6 Triangular representations of operators close to unitary 7 LU-factorization of semi-separable integral operators 8 Generalised Wiener-Hopf equations 9 Generalised LU-factorization relative to discrete chains Comments on Part V Exercises to Part V PART VI: CLASSES OF TOEPLITZ OPERATORS XXIII Block Toeplitz operators 1 Preliminaries 2 Block Laurent operators 3 Block Toeplitz operators 4 Block Toeplitz operators defined by continuous functions 5 The Fredholm index of a block Toeplitz operator defined by a continuous function XXIV Toeplitz operators defined by rational matrix functions 1 Preliminaries 2 Invertibility and Fredholm index (scalar case) 3 Wiener-Hopf factorization 4 Invertibility and Fredholm index (matrix case) 5 Intermezzo about realisation 6 Inversion of a block Laurent operator 7 Explicit canonical factorization 8 Explicit inversion formulas 9 Explicit formulas for Fredholm characteristics 10 An example 11 Asymptotic formulas for determinants of block Toeplitz matrices XXV Toeplitz operators defined by piecewise continuous matrix functions 1 Piecewise continuous functions 2 Symbol and Fredholm index (scalar case) 3 Symbol and Fredholm index (matrix case) 4 Sums of products of Toeplitz operators defined by piecewise continuous functions 5 Sums of products of block Toeplitz operators defined by piecewise continuous functions Comments on Part VI Exercises to Part VI PART VII: CONTRACTIVE OPERATORS AND CHARACTERISTIC OPERATOR FUNCTIONS XXVI Block shift operators 1 Forward shifts and isometries 2 Parts of block shift operators 3 Invariant subspaces of forward shift operators XXVII Dilation theory 1 Preliminaries about contractions 2 Preliminaries about dilations 3 Isometric dilations 4 Unitary dilations

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979 citations