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Approximation Of Continuously Differentiable Functions
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Book Synopsis Approximation of Continuously Differentiable Functions by : J.G. Llavona
Download or read book Approximation of Continuously Differentiable Functions written by J.G. Llavona and published by Elsevier. This book was released on 1986-11-01 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: This self-contained book brings together the important results of a rapidly growing area.As a starting point it presents the classic results of the theory. The book covers such results as: the extension of Wells' theorem and Aron's theorem for the fine topology of order m; extension of Bernstein's and Weierstrass' theorems for infinite dimensional Banach spaces; extension of Nachbin's and Whitney's theorem for infinite dimensional Banach spaces; automatic continuity of homomorphisms in algebras of continuously differentiable functions, etc.
Book Synopsis Tractability of Multivariate Problems: Linear information by : Erich Novak
Download or read book Tractability of Multivariate Problems: Linear information written by Erich Novak and published by European Mathematical Society. This book was released on 2008 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multivariate problems occur in many applications. These problems are defined on spaces of $d$-variate functions and $d$ can be huge--in the hundreds or even in the thousands. Some high-dimensional problems can be solved efficiently to within $\varepsilon$, i.e., the cost increases polynomially in $\varepsilon^{-1}$ and $d$. However, there are many multivariate problems for which even the minimal cost increases exponentially in $d$. This exponential dependence on $d$ is called intractability or the curse of dimensionality. This is the first volume of a three-volume set comprising a comprehensive study of the tractability of multivariate problems. It is devoted to tractability in the case of algorithms using linear information and develops the theory for multivariate problems in various settings: worst case, average case, randomized and probabilistic. A problem is tractable if its minimal cost is not exponential in $\varepsilon^{-1}$ and $d$. There are various notions of tractability, depending on how we measure the lack of exponential dependence. For example, a problem is polynomially tractable if its minimal cost is polynomial in $\varepsilon^{-1}$ and $d$. The study of tractability was initiated about 15 years ago. This is the first and only research monograph on this subject. Many multivariate problems suffer from the curse of dimensionality when they are defined over classical (unweighted) spaces. In this case, all variables and groups of variables play the same role, which causes the minimal cost to be exponential in $d$. But many practically important problems are solved today for huge $d$ in a reasonable time. One of the most intriguing challenges of the theory is to understand why this is possible. Multivariate problems may become weakly tractable, polynomially tractable or even strongly polynomially tractable if they are defined over weighted spaces with properly decaying weights. One of the main purposes of this book is to study weighted spaces and obtain necessary and sufficient conditions on weights for various notions of tractability. The book is of interest for researchers working in computational mathematics, especially in approximation of high-dimensional problems. It may be also suitable for graduate courses and seminars. The text concludes with a list of thirty open problems that can be good candidates for future tractability research.
Author :Vladimir I. Arnold Publisher :Springer Science & Business Media ISBN 13 :9783540548133 Total Pages :346 pages Book Rating :4.5/5 (481 download)
Book Synopsis Ordinary Differential Equations by : Vladimir I. Arnold
Download or read book Ordinary Differential Equations written by Vladimir I. Arnold and published by Springer Science & Business Media. This book was released on 1992-05-08 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: Few books on Ordinary Differential Equations (ODEs) have the elegant geometric insight of this one, which puts emphasis on the qualitative and geometric properties of ODEs and their solutions, rather than on routine presentation of algorithms. From the reviews: "Professor Arnold has expanded his classic book to include new material on exponential growth, predator-prey, the pendulum, impulse response, symmetry groups and group actions, perturbation and bifurcation." --SIAM REVIEW
Book Synopsis The Approximation of Continuous Functions by Positive Linear Operators by : Ronald A. De Vore
Download or read book The Approximation of Continuous Functions by Positive Linear Operators written by Ronald A. De Vore and published by Springer. This book was released on 2006-11-15 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Topics in Uniform Approximation of Continuous Functions by : Ileana Bucur
Download or read book Topics in Uniform Approximation of Continuous Functions written by Ileana Bucur and published by Springer Nature. This book was released on 2020-08-18 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the evolution of uniform approximations of continuous functions. Starting from the simple case of a real continuous function defined on a closed real interval, i.e., the Weierstrass approximation theorems, it proceeds up to the abstract case of approximation theorems in a locally convex lattice of (M) type. The most important generalizations of Weierstrass’ theorems obtained by Korovkin, Bohman, Stone, Bishop, and Von Neumann are also included. In turn, the book presents the approximation of continuous functions defined on a locally compact space (the functions from a weighted space) and that of continuous differentiable functions defined on ¡n. In closing, it highlights selected approximation theorems in locally convex lattices of (M) type. The book is intended for advanced and graduate students of mathematics, and can also serve as a resource for researchers in the field of the theory of functions.
Author :Hans-Jürgen Reinhardt Publisher :Springer Science & Business Media ISBN 13 :1461210801 Total Pages :412 pages Book Rating :4.4/5 (612 download)
Book Synopsis Analysis of Approximation Methods for Differential and Integral Equations by : Hans-Jürgen Reinhardt
Download or read book Analysis of Approximation Methods for Differential and Integral Equations written by Hans-Jürgen Reinhardt and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 412 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is primarily based on the research done by the Numerical Analysis Group at the Goethe-Universitat in Frankfurt/Main, and on material presented in several graduate courses by the author between 1977 and 1981. It is hoped that the text will be useful for graduate students and for scientists interested in studying a fundamental theoretical analysis of numerical methods along with its application to the most diverse classes of differential and integral equations. The text treats numerous methods for approximating solutions of three classes of problems: (elliptic) boundary-value problems, (hyperbolic and parabolic) initial value problems in partial differential equations, and integral equations of the second kind. The aim is to develop a unifying convergence theory, and thereby prove the convergence of, as well as provide error estimates for, the approximations generated by specific numerical methods. The schemes for numerically solving boundary-value problems are additionally divided into the two categories of finite difference methods and of projection methods for approximating their variational formulations.
Book Synopsis Implicit Functions and Solution Mappings by : Asen L. Dontchev
Download or read book Implicit Functions and Solution Mappings written by Asen L. Dontchev and published by Springer. This book was released on 2014-06-18 with total page 495 pages. Available in PDF, EPUB and Kindle. Book excerpt: The implicit function theorem is one of the most important theorems in analysis and its many variants are basic tools in partial differential equations and numerical analysis. This second edition of Implicit Functions and Solution Mappings presents an updated and more complete picture of the field by including solutions of problems that have been solved since the first edition was published, and places old and new results in a broader perspective. The purpose of this self-contained work is to provide a reference on the topic and to provide a unified collection of a number of results which are currently scattered throughout the literature. Updates to this edition include new sections in almost all chapters, new exercises and examples, updated commentaries to chapters and an enlarged index and references section.
Author :Nikolaĭ Pavlovich Korneĭchuk Publisher :Cambridge University Press ISBN 13 :9780521382342 Total Pages :472 pages Book Rating :4.3/5 (823 download)
Book Synopsis Exact Constants in Approximation Theory by : Nikolaĭ Pavlovich Korneĭchuk
Download or read book Exact Constants in Approximation Theory written by Nikolaĭ Pavlovich Korneĭchuk and published by Cambridge University Press. This book was released on 1991-06-06 with total page 472 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended as a self-contained introduction for non-specialists, or as a reference work for experts, to the particular area of approximation theory that is concerned with exact constants. The results apply mainly to extremal problems in approximation theory, which in turn are closely related to numerical analysis and optimization. The book encompasses a wide range of questions and problems: best approximation by polynomials and splines; linear approximation methods, such as spline-approximation; optimal reconstruction of functions and linear functionals. Many of the results are based on deep facts from analysis and function theory, such as duality theory and comparison theorems; these are presented in chapters 1 and 3. In keeping with the author's intention to make the book as self-contained as possible, chapter 2 contains an introduction to polynomial and spline approximation. Chapters 4 to 7 apply the theory to specific classes of functions. The last chapter deals with n-widths and generalises some of the ideas of the earlier chapters. Each chapter concludes with commentary, exercises and extensions of results. A substantial bibliography is included. Many of the results collected here have not been gathered together in book form before, so it will be essential reading for approximation theorists.
Book Synopsis Approximation Theory and Approximation Practice, Extended Edition by : Lloyd N. Trefethen
Download or read book Approximation Theory and Approximation Practice, Extended Edition written by Lloyd N. Trefethen and published by SIAM. This book was released on 2019-01-01 with total page 377 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a textbook on classical polynomial and rational approximation theory for the twenty-first century. Aimed at advanced undergraduates and graduate students across all of applied mathematics, it uses MATLAB to teach the fields most important ideas and results. Approximation Theory and Approximation Practice, Extended Edition differs fundamentally from other works on approximation theory in a number of ways: its emphasis is on topics close to numerical algorithms; concepts are illustrated with Chebfun; and each chapter is a PUBLISHable MATLAB M-file, available online. The book centers on theorems and methods for analytic functions, which appear so often in applications, rather than on functions at the edge of discontinuity with their seductive theoretical challenges. Original sources are cited rather than textbooks, and each item in the bibliography is accompanied by an editorial comment. In addition, each chapter has a collection of exercises, which span a wide range from mathematical theory to Chebfun-based numerical experimentation. This textbook is appropriate for advanced undergraduate or graduate students who have an understanding of numerical analysis and complex analysis. It is also appropriate for seasoned mathematicians who use MATLAB.
Book Synopsis Canadian Journal of Mathematics by :
Download or read book Canadian Journal of Mathematics written by and published by . This book was released on 1989-02 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Course in Ordinary and Partial Differential Equations by : Zalman Rubinstein
Download or read book A Course in Ordinary and Partial Differential Equations written by Zalman Rubinstein and published by Academic Press. This book was released on 2014-05-12 with total page 488 pages. Available in PDF, EPUB and Kindle. Book excerpt: A Course in Ordinary and Partial Differential Equations discusses ordinary differential equations and partial differential equations. The book reviews the solution of elementary first-order differential equations, existence theorems, singular solutions, and linear equations of arbitrary order. It explains the solutions of linear equations with constant coefficients, operational calculus, and the solutions of linear differential equations. It also explores the techniques of computing for the solution of systems of linear differential equations, which is similar to the solutions of linear equations of arbitrary order. The text proves that if the coefficients of some differential equations possess certain restricted types of singularities, the solution will have Taylor series expansions about the singular points. The investigator can calculate a divergent series whose partial sums numerically approximate the solution for large x if the point in question is infinity, of which the series will be a Taylor series of negative powers of x. The book also explains the Fourier transform, its applications to partial differential equations, as well as the Hilbert space approach to partial differential equations. The book is a stimulating material for mathematicians, for professors, or for students of pure and applied mathematics, physics, or engineering.
Book Synopsis Theory and Applications of Differentiable Functions of Several Variables by :
Download or read book Theory and Applications of Differentiable Functions of Several Variables written by and published by American Mathematical Soc.. This book was released on 1994 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection is the 14th in an ongoing series on differentiable functions of several variables, presenting recent contributions to a line of research begun by Sobolev in 1950. The papers study various spaces of differentiable functions of several real variables in Euclidean space, their imbeddings, equivalent normings, weighted estimates of derivatives, and traces on sets. Several questions of approximation in function spaces on the line, on a hyperboloid, and on Lobachevsky space are studied. Investigations of bilinear approximations are applied to estimates of the singular numbers of integral operators and widths. The authors also examine the asymptotics of the spectrum of elliptic systems, as well as the Dirichlet variational problem for a degenerate elliptic operator. Finally, a block method of solving Laplace's equation for nonanalytic boundary conditions is developed.
Book Synopsis Computed Tomography by : Per Christian Hansen
Download or read book Computed Tomography written by Per Christian Hansen and published by SIAM. This book was released on 2021-09-25 with total page 355 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes fundamental computational methods for image reconstruction in computed tomography (CT) with a focus on a pedagogical presentation of these methods and their underlying concepts. Insights into the advantages, limitations, and theoretical and computational aspects of the methods are included, giving a balanced presentation that allows readers to understand and implement CT reconstruction algorithms. Unique in its emphasis on the interplay between modeling, computing, and algorithm development, Computed Tomography: Algorithms, Insight, and Just Enough Theory develops the mathematical and computational aspects of three main classes of reconstruction methods: classical filtered back-projection, algebraic iterative methods, and variational methods based on nonlinear numerical optimization algorithms. It spotlights the link between CT and numerical methods, which is rarely discussed in current literature, and describes the effects of incomplete data using both microlocal analysis and singular value decomposition (SVD). This book sets the stage for further exploration of CT algorithms. Readers will be able to grasp the underlying mathematical models to motivate and derive the basic principles of CT reconstruction and will gain basic understanding of fundamental computational challenges of CT, such as the influence of noisy and incomplete data, as well as the reconstruction capabilities and the convergence of the iterative algorithms. Exercises using MATLAB are included, allowing readers to experiment with the algorithms and making the book suitable for teaching and self-study. Computed Tomography: Algorithms, Insight, and Just Enough Theory is primarily aimed at students, researchers, and practitioners interested in the computational aspects of X-ray CT and is also relevant for anyone working with other forms of tomography, such as neutron and electron tomography, that share the same mathematical formulation. With its basis in lecture notes developed for a PhD course, it is appropriate as a textbook for courses on computational methods for X-ray CT and computational methods for inverse problems.
Book Synopsis Approximation and Optimization of Discrete and Differential Inclusions by : Elimhan N Mahmudov
Download or read book Approximation and Optimization of Discrete and Differential Inclusions written by Elimhan N Mahmudov and published by Elsevier. This book was released on 2011-08-25 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: Optimal control theory has numerous applications in both science and engineering. This book presents basic concepts and principles of mathematical programming in terms of set-valued analysis and develops a comprehensive optimality theory of problems described by ordinary and partial differential inclusions. In addition to including well-recognized results of variational analysis and optimization, the book includes a number of new and important ones Includes practical examples
Book Synopsis World Congress of Nonlinear Analysts '92 by : V. Lakshmikantham
Download or read book World Congress of Nonlinear Analysts '92 written by V. Lakshmikantham and published by Walter de Gruyter. This book was released on 2011-11-14 with total page 4040 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Optimization of Resonances for Multilayer X-ray Resonators by : Felix Schenk
Download or read book Optimization of Resonances for Multilayer X-ray Resonators written by Felix Schenk and published by Universitätsverlag Göttingen. This book was released on 2011 with total page 134 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Applied Production Analysis by : Robert G. Chambers
Download or read book Applied Production Analysis written by Robert G. Chambers and published by Cambridge University Press. This book was released on 1988-07-29 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains a modern treatment of production economics from a dual perspective, with special emphasis on recent developments. Results that were scattered throughout professional journals and monographs are now gathered into an integrated approach using a common notation. The book prepares the reader to apply the tools of the dual approach to real world problems and data sets. Particular care has been devoted to choosing topics for discussion that achieve this goal. Throughout the book there are worked examples and exercises, which are geared toward developing the reader's facility in using modern developments in production economics. Separate chapters are devoted to production, cost, and profit functions. Other topics include flexible functional forms, aggregation across inputs and outputs using the theory of separable structures, aggregation over economic optimizing firms, the representation of multioutput technologies, and the analysis and measurement of technical change from both a primal and a dual perspective.
