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The theory of operator spaces is very recent and can be described as a non-commutative Banach space theory. An 'operator space' is simply a Banach space with an embedding into the space B(H) of all bounded operators on a Hilbert space H. The first part of this book is an introduction with emphasis on examples that illustrate various aspects of the theory. The second part is devoted to applications to C*-algebras, with a systematic exposition of tensor products of C*-algebras. The third (and shorter) part of the book describes applications to non self-adjoint operator algebras, and similarity problems. In particular the author's counterexample to the 'Halmos problem' is presented, as well as work on the new concept of 'length' of an operator algebra. Graduate students and professional mathematicians interested in functional analysis, operator algebras and theoretical physics will find that this book has much to offer.

Operator spaces. --- Matricially normed spaces --- Non-commutative Banach space theory --- Banach spaces --- Operator theory --- Operator spaces

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Regularization methods aimed at finding stable approximate solutions are a necessary tool to tackle inverse and ill-posed problems. Inverse problems arise in a large variety of applications ranging from medical imaging and non-destructive testing via finance to systems biology. Many of these problems belong to the class of parameter identification problems in partial differential equations (PDEs) and thus are computationally demanding and mathematically challenging. Hence there is a substantial need for stable and efficient solvers for this kind of problems as well as for a rigorous convergence analysis of these methods. This monograph consists of five parts. Part I motivates the importance of developing and analyzing regularization methods in Banach spaces by presenting four applications which intrinsically demand for a Banach space setting and giving a brief glimpse of sparsity constraints. Part II summarizes all mathematical tools that are necessary to carry out an analysis in Banach spaces. Part III represents the current state-of-the-art concerning Tikhonov regularization in Banach spaces. Part IV about iterative regularization methods is concerned with linear operator equations and the iterative solution of nonlinear operator equations by gradient type methods and the iteratively regularized Gauß-Newton method. Part V finally outlines the method of approximate inverse which is based on the efficient evaluation of the measured data with reconstruction kernels.

Banach spaces --- Parameter estimation --- Differential equations, Partial --- Banach, Espaces de --- Estimation d'un paramètre --- Equations aux dérivées partielles --- Banach spaces. --- Parameter estimation. --- Differential equations, Partial. --- Partial differential equations --- Estimation theory --- Stochastic systems --- Functions of complex variables --- Generalized spaces --- Topology --- Banach Space. --- Iterative Method. --- Regularization Theory. --- Tikhonov Regularization.

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A plethora of problems from diverse disciplines such as Mathematics, Mathematical: Biology, Chemistry, Economics, Physics, Scientific Computing and also Engineering can be formulated as an equation defined in abstract spaces using Mathematical Modelling. The solutions of these equations can be found in closed form only in special case. That is why researchers and practitioners utilize iterative procedures from which a sequence is being generated approximating the solution under some conditions on the initial data. This type of research is considered most interesting and challenging. This is our motivation for the introduction of this special issue on Iterative Procedures.

Lipschitz condition --- order of convergence --- Scalar equations --- local and semilocal convergence --- multiple roots --- Nondifferentiable operator --- optimal iterative methods --- Order of convergence --- convergence order --- fast algorithms --- iterative method --- computational convergence order --- generalized mixed equilibrium problem --- nonlinear equations --- systems of nonlinear equations --- Chebyshev’s iterative method --- local convergence --- iterative methods --- divided difference --- Multiple roots --- semi-local convergence --- scalar equations --- left Bregman asymptotically nonexpansive mapping --- basin of attraction --- maximal monotone operator --- Newton–HSS method --- general means --- Steffensen’s method --- derivative-free method --- simple roots --- fixed point problem --- split variational inclusion problem --- weighted-Newton method --- ball radius of convergence --- Traub–Steffensen method --- Newton’s method --- fractional derivative --- Banach space --- multiple-root solvers --- uniformly convex and uniformly smooth Banach space --- Fréchet-derivative --- optimal convergence --- Optimal iterative methods --- basins of attraction --- nonlinear equation

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Embeddings of discrete metric spaces into Banach spaces recently became an important tool in computer science and topology. The purpose of the book is to present some of the most important techniques and results, mostly on bilipschitz and coarse embeddings. The topics include: (1) Embeddability of locally finite metric spaces into Banach spaces is finitely determined; (2) Constructions of embeddings; (3) Distortion in terms of Poincaré inequalities; (4) Constructions of families of expanders and of families of graphs with unbounded girth and lower bounds on average degrees; (5) Banach spaces which do not admit coarse embeddings of expanders; (6) Structure of metric spaces which are not coarsely embeddable into a Hilbert space; (7) Applications of Markov chains to embeddability problems; (8) Metric characterizations of properties of Banach spaces; (9) Lipschitz free spaces. Substantial part of the book is devoted to a detailed presentation of relevant results of Banach space theory and graph theory. The final chapter contains a list of open problems. Extensive bibliography is also included. Each chapter, except the open problems chapter, contains exercises and a notes and remarks section containing references, discussion of related results, and suggestions for further reading. The book will help readers to enter and to work in a very rapidly developing area having many important connections with different parts of mathematics and computer science.

Banach spaces. --- Lipschitz spaces. --- Stochastic partial differential equations. --- Banach spaces, Stochastic differential equations in --- Hilbert spaces, Stochastic differential equations in --- SPDE (Differential equations) --- Stochastic differential equations in Banach spaces --- Stochastic differential equations in Hilbert spaces --- Differential equations, Partial --- Hölder spaces --- Function spaces --- Functions of complex variables --- Generalized spaces --- Topology --- Banach Space Theory. --- Bilipschitz Embedding. --- Coarse Embedding. --- Embedding of Discrete Metric Spaces. --- Functional Analysis. --- Graph Theory.

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In this book the authors give the first necessary and sufficient conditions for the uniform convergence a.s. of random Fourier series on locally compact Abelian groups and on compact non-Abelian groups. They also obtain many related results. For example, whenever a random Fourier series converges uniformly a.s. it also satisfies the central limit theorem. The methods developed are used to study some questions in harmonic analysis that are not intrinsically random. For example, a new characterization of Sidon sets is derived.The major results depend heavily on the Dudley-Fernique necessary and sufficient condition for the continuity of stationary Gaussian processes and on recent work on sums of independent Banach space valued random variables. It is noteworthy that the proofs for the Abelian case immediately extend to the non-Abelian case once the proper definition of random Fourier series is made. In doing this the authors obtain new results on sums of independent random matrices with elements in a Banach space. The final chapter of the book suggests several directions for further research.

Harmonic analysis. Fourier analysis --- Fourier series. --- Harmonic analysis. --- Fourier, Séries de --- Analyse harmonique --- 517.518.4 --- Fourier series --- Harmonic analysis --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Harmonic functions --- Time-series analysis --- Fourier integrals --- Series, Fourier --- Series, Trigonometric --- Trigonometric series --- Fourier analysis --- 517.518.4 Trigonometric series --- Fourier, Séries de --- Abelian group. --- Almost periodic function. --- Almost surely. --- Banach space. --- Big O notation. --- Cardinality. --- Central limit theorem. --- Circle group. --- Coefficient. --- Commutative property. --- Compact group. --- Compact space. --- Complex number. --- Continuous function. --- Corollary. --- Discrete group. --- Equivalence class. --- Existential quantification. --- Finite group. --- Gaussian process. --- Haar measure. --- Independence (probability theory). --- Inequality (mathematics). --- Integer. --- Irreducible representation. --- Non-abelian group. --- Non-abelian. --- Normal distribution. --- Orthogonal group. --- Orthogonal matrix. --- Probability distribution. --- Probability measure. --- Probability space. --- Probability. --- Random function. --- Random matrix. --- Random variable. --- Rate of convergence. --- Real number. --- Ring (mathematics). --- Scientific notation. --- Set (mathematics). --- Slepian's lemma. --- Small number. --- Smoothness. --- Stationary process. --- Subgroup. --- Subset. --- Summation. --- Theorem. --- Uniform convergence. --- Unitary matrix. --- Variance.