Toeplitz matrix

About: Toeplitz matrix is a research topic. Over the lifetime, 8097 publications have been published within this topic receiving 131901 citations.

Classes of Linear Operators

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

Analysis of Toeplitz Operators

Abstract: Toeplitz operators arise in plenty of applications. They constitute one of the most important classes of non-selfadjoint operators, and the ideas and methods prevailing in the field of Toeplitz operators are a fascinating illustration of the fruitful interplay between operator theory, complex analysis, and Banach algebra techniques. This book is a systematic introduction to the advanced analysis of block Toeplitz operators and includes both classical results and recent developments. Its first edition has been a standard reference for fifteen years. The present second edition is enriched by several results obtained only in the last decade. The topics treated range from the analysis of locally sectorial matrix functions through Toeplitz and Wiener-Hopf operators on Banach spaces, projection methods, and quarter-plane operators up to Toeplitz and Wiener-Hopf determinants. The book is addressed to both graduate students approaching the subject for the first time and specialists in the theory of Toeplitz operators, but should also be of interest to physicists, probabilists, and computer scientists.

Efficient parameter estimation for self-similar processes

Abstract: Asymptotic normality of the maximum likelihood estimator for the parameters of a long range dependent Gaussian process is proved. Furthermore, the limit of the Fisher information matrix is derived for such processes which implies efficiency of the estimator and of an approximate maximum likelihood estimator studied by Fox and Taqqu. The results are derived by using asymptotic properties of Toeplitz matrices and an equicontinuity property of quadratic forms.

Conjugate Gradient Methods for Toeplitz Systems

Abstract: In this expository paper, we survey some of the latest developments in using preconditioned conjugate gradient methods for solving Toeplitz systems. One of the main results is that the complexity of solving a large class of $n$-by-$n$ Toeplitz systems is reduced to $O(n \log n)$ operations as compared to $O(n \log ^2 n)$ operations required by fast direct Toeplitz solvers. Different preconditioners proposed for Toeplitz systems are reviewed. Applications to Toeplitz-related systems arising from partial differential equations, queueing networks, signal and image processing, integral equations, and time series analysis are given.