Operators and Iterative Processes of Fejer Type pdf epub mobi txt 电子书 下载 2026

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出版者:

作者:Vasin, Vladimir V./ Eremin, Ivan I.

出品人:

页数:155

译者:

出版时间:

价格:1221.00元

装帧:

isbn号码:9783110218183

丛书系列:

图书标签:

• Fejer type operators
• Iterative processes
• Approximation theory
• Numerical analysis
• Functional analysis
• Operator theory
• Mathematical analysis
• Real analysis
• Fixed point theory
• Convergence analysis

Operator Theory and Abstract Analysis: A Deep Dive into Iterative Processes This monograph offers a comprehensive exploration of operator theory, focusing on the intricate interplay between linear operators and the iterative processes they generate. The work delves into the foundational principles that underpin the behavior of operators within abstract spaces and examines how these operators can be harnessed to construct and analyze powerful iterative schemes. The primary objective is to provide a rigorous and insightful account of the theoretical underpinnings of these iterative methods, with a particular emphasis on their convergence properties, stability, and applications in various fields of mathematics and science. The book is structured to guide the reader from fundamental concepts to advanced topics, ensuring a solid grasp of the material at each stage. We begin by establishing a robust framework for understanding linear operators. This includes a detailed discussion of operator norms, spectral properties, and the relationship between operators and the topological and geometric structures of the spaces they act upon. Concepts such as Banach spaces, Hilbert spaces, and their associated properties are introduced and explored in depth. The spectral theory of operators, a cornerstone of functional analysis, is presented with careful attention to detail, including discussions on eigenvalues, eigenvectors, and the spectral radius. This foundational knowledge is crucial for comprehending the dynamics of iterative processes. A significant portion of the book is dedicated to the theory of iterative methods. We begin with an examination of simple iterative schemes, such as those derived from fixed-point theorems. This includes a thorough analysis of the conditions under which these methods converge, the rates of convergence, and the factors that influence their stability. The role of operator properties, such as contractivity, monotonicity, and self-adjointness, in guaranteeing convergence is investigated extensively. We explore classical iterative methods like the Jacobi method, Gauss-Seidel method, and successive over-relaxation (SOR) method in the context of solving linear systems, demonstrating how their convergence behavior can be analyzed using operator theoretic tools. The monograph then extends its focus to more sophisticated iterative processes, including those arising in the context of nonlinear operators. The theory of Newton's method and its variants is presented, highlighting its quadratic convergence under suitable conditions. The challenges associated with applying iterative methods to nonlinear problems, such as the existence and uniqueness of solutions and the robustness of the iterative schemes, are discussed in detail. We investigate how topological properties of the underlying spaces and the differentiability of the operators influence the behavior of these nonlinear iterative processes. A key theme woven throughout the text is the profound connection between Fejér-type processes and operator theory. Fejér’s theorem, originally concerning trigonometric polynomials, has inspired a broad class of iterative algorithms that exhibit remarkable convergence properties, often even in the absence of strict convexity or other strong assumptions. This book provides a comprehensive theoretical framework for understanding these Fejér-type iterations. We explore their construction from fundamental operator-theoretic principles, such as projections and averaging operators. The generalization of Fejér's original ideas to infinite-dimensional spaces and to a wider range of optimization and signal processing problems is a central focus. The book elucidates how these processes can be viewed as projections onto convex sets or as sequences of approximations that progressively refine a solution. The analysis of Fejér-type processes involves delving into the geometric interpretation of operators and their action on sets. Concepts like convex hulls, projections onto convex sets, and the characterization of fixed points of nonexpansive operators are explored and linked to the convergence of these iterative schemes. We investigate specific classes of operators that naturally give rise to Fejér-type iterations, such as those arising in inverse problems, signal recovery, and convex feasibility problems. The book provides detailed proofs and derivations for the convergence theorems associated with these processes, emphasizing the role of monotonicity, demi-closedness, and other crucial operator properties. Furthermore, the monograph addresses the practical aspects of implementing and analyzing iterative processes. This includes discussions on numerical stability, the choice of appropriate parameters, and techniques for accelerating convergence. We also explore the relationship between abstract operator theory and its concrete applications in areas such as numerical linear algebra, partial differential equations, and optimization. The book showcases how the theoretical tools developed can be used to analyze the efficiency and reliability of algorithms employed in solving real-world problems. The theoretical apparatus employed draws heavily from functional analysis, including measure theory, integration, and the theory of function spaces. However, the exposition is crafted to be accessible to readers with a solid background in real analysis and linear algebra, with necessary background material and definitions provided within the text or through clear references. The book aims to foster a deep understanding of the underlying mathematical principles, rather than merely presenting a collection of algorithms. To facilitate comprehension, the book includes numerous examples and case studies that illustrate the application of the theoretical concepts. These examples range from classic problems in numerical analysis to contemporary challenges in data science and machine learning, where iterative methods play a pivotal role. The text also highlights open problems and directions for future research, encouraging readers to engage with the ongoing development of operator theory and iterative processes. In summary, this monograph provides a rigorous and in-depth treatment of operator theory with a strong emphasis on the development and analysis of iterative processes, particularly those of Fejér type. It bridges the gap between abstract mathematical theory and practical computational methods, offering a valuable resource for researchers and graduate students in mathematics, computer science, engineering, and related fields. The detailed exploration of convergence properties, stability, and applications ensures that readers gain a comprehensive understanding of the power and elegance of operator-theoretic approaches to solving complex problems.

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