This work deals with the many variations of the Stoneileierstrass Theorem for vector-valued functions and some of its applications. The book is largely self-contained.
This work deals with the many variations of the Stoneileierstrass Theorem for vector-valued functions and some of its applications. The book is largely self-contained. The amount of Functional Analysis required is minimal, except for Chapter 8. The book can be used by graduate students who have taken the usual first-year real and complex analysis courses.
Bacopoulos, A., Godini, G., Singer, I., “On best approximation in vector-valued norms”, Colloq. ... I., “On infima of sets in the plane and best approximation,
simultaneous and vectorial, in a linear space with two norms”, in: Frehse, J.,
Author: Johannes Jahn
Publisher: Springer Science & Business Media
Category: Business & Economics
In vector optimization one investigates optimal elements such as min imal, strongly minimal, properly minimal or weakly minimal elements of a nonempty subset of a partially ordered linear space. The prob lem of determining at least one of these optimal elements, if they exist at all, is also called a vector optimization problem. Problems of this type can be found not only in mathematics but also in engineer ing and economics. Vector optimization problems arise, for exam ple, in functional analysis (the Hahn-Banach theorem, the lemma of Bishop-Phelps, Ekeland's variational principle), multiobjective pro gramming, multi-criteria decision making, statistics (Bayes solutions, theory of tests, minimal covariance matrices), approximation theory (location theory, simultaneous approximation, solution of boundary value problems) and cooperative game theory (cooperative n player differential games and, as a special case, optimal control problems). In the last decade vector optimization has been extended to problems with set-valued maps. This new field of research, called set optimiza tion, seems to have important applications to variational inequalities and optimization problems with multivalued data. The roots of vector optimization go back to F. Y. Edgeworth (1881) and V. Pareto (1896) who has already given the definition of the standard optimality concept in multiobjective optimization. But in mathematics this branch of optimization has started with the leg endary paper of H. W. Kuhn and A. W. Tucker (1951). Since about v Vl Preface the end of the 60's research is intensively made in vector optimization.
There is a difference, however, since in (3.4) the Hilbert space norm || || is used
since we are dealing with a vector valued kernel. In order to apply transference
methods to this more general situation we can use the following result: (3.6) The
The publication of Oberwolfach conference books was initiated by Birkhauser Publishers in 1964 with the proceedings of the conference 'On Approximation Theory', conducted by P. L. Butzer (Aachen) and J. Korevaar (Amsterdam). Since that auspicious beginning, others of the Oberwolfach proceedings have appeared in Birkhauser's ISNM series. The present volume is the fifth * edited at Aachen in collaboration with an external institution. It once again ad dresses itself to the most recent results on approximation and operator theory, and includes 47 of the 48 lectures presented at Oberwolfach, as well as five articles subsequently submitted by V. A. Baskakov (Moscow), H. Esser (Aachen), G. Lumer (Mons), E. L. Stark (Aachen) and P. M. Tamrazov (Kiev). In addition, there is a section devoted to new and unsolved problems, based upon two special problem sessions augmented by later communications from the participants. Corresponding to the nature of the conference, the aim of the organizers was to solicit both specialized and survey papers, ranging in the broad area of classical and functional analysis, from approximation and interpolation theory to Fourier and harmonic analysis, and to the theory of function spaces and operators. The papers were supplemented by lectures on fields represented for the first time in our series of Oberwolfach Conferences, so for example, complex function theory or probability and sampling theory.
Vector-valued Functions. ... Let B be a compact set and let C,(B) be the linear space of all m-dimensional vectors f=(f1, f2, ..., f,) whose components are ... (3.19)
The following theorems hold for approximation in the sense of the norm (3.19).
Author: Günter Meinardus
Publisher: Springer Science & Business Media
Category: Juvenile Nonfiction
for example, the so-called Lp approximation, the Bernstein approxima tion problem (approximation on the real line by certain entire functions), and the highly interesting studies of J. L. WALSH on approximation in the complex plane. I would like to extend sincere thanks to Professor L. COLLATZ for his many encouragements for the writing of this book. Thanks are equally due to Springer-Verlag for their ready agreement to my wishes, and for the excellent and competent composition of the book. In addition, I would like to thank Dr. W. KRABS, Dr. A. -G. MEYER and D. SCHWEDT for their very careful reading of the manuscript. Hamburg, March 1964 GUNTER MEINARDUS Preface to the English Edition This English edition was translated by Dr. LARRY SCHUMAKER, Mathematics Research Center, United States Army, The University of Wisconsin, Madison, from a supplemented version of the German edition. Apart from a number of minor additions and corrections and a few new proofs (e. g. , the new proof of JACKSON'S Theorem), it differs in detail from the first edition by the inclusion of a discussion of new work on comparison theorems in the case of so-called regular Haar systems (§ 6) and on Segment Approximation (§ 11). I want to thank the many readers who provided comments and helpful suggestions. My special thanks are due to the translator, to Springer-Verlag for their ready compliance with all my wishes, to Mr.
9 Vector Approximation Vector approximation problems are abstract approximation problems where a vectorial norm is used instead of a usual ( real - valued ) norm . Many important results known from approximation theory can be
Author: Johannes Jahn
Publisher: Peter Lang Gmbh, Internationaler Verlag Der Wissenschaften
Category: Linear topological spaces
In vector optimization one investigates optimal elements such as minimal, strongly minimal, properly minimal or weakly minimal elements of a nonempty subset of a partially ordered linear space. The problem of determining at least one of these optimal elements, if they exist at all, is also called a vector optimization problem. Problems of this type can be found not only in mathematics but also in engineering and economics. Vector optimization problems arise, for example, in functional analysis (the Hahn-Banach theorem, the lemma of Bishop-Phelps, Ekeland's variational principle), multi-objective programming, multi-criteria decision making, statistics (Bayes solutions, theory of tests, minimal covariance matrices), approximation theory (location theory, simultaneous approximation, solution of boundary value problems) and cooperative game theory (cooperative n player differential games and, as a special case, optimal control problems).
the " distance of a point to a subset of a linear space endowed with a ( vector - valued ) norm with values in RP . ... normed linear space , G a non - void subset
of E and se € E . An element go e G is called an element of best approximation of
A normed vector space is an ordinary vector space equipped with a norm, a well-
behaved function that assigns a nonnegative "length" to each vector. Basic
topological concepts ... The chapter highlights the idea of approximation through
mean value theorems, which allow us to estimate the increments of a function in
terms of increments of the approximating linear transformation. In Chapter 6 we
Author: Mustafa A. Akcoglu
Publisher: John Wiley & Sons
A rigorous introduction to calculus in vector spaces The concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. Analysis in Vector Spaces presents the central results of this classic subject through rigorous arguments, discussions, and examples. The book aims to cultivate not only knowledge of the major theoretical results, but also the geometric intuition needed for both mathematical problem-solving and modeling in the formal sciences. The authors begin with an outline of key concepts, terminology, and notation and also provide a basic introduction to set theory, the properties of real numbers, and a review of linear algebra. An elegant approach to eigenvector problems and the spectral theorem sets the stage for later results on volume and integration. Subsequent chapters present the major results of differential and integral calculus of several variables as well as the theory of manifolds. Additional topical coverage includes: Sets and functions Real numbers Vector functions Normed vector spaces First- and higher-order derivatives Diffeomorphisms and manifolds Multiple integrals Integration on manifolds Stokes' theorem Basic point set topology Numerous examples and exercises are provided in each chapter to reinforce new concepts and to illustrate how results can be applied to additional problems. Furthermore, proofs and examples are presented in a clear style that emphasizes the underlying intuitive ideas. Counterexamples are provided throughout the book to warn against possible mistakes, and extensive appendices outline the construction of real numbers, include a fundamental result about dimension, and present general results about determinants. Assuming only a fundamental understanding of linear algebra and single variable calculus, Analysis in Vector Spaces is an excellent book for a second course in analysis for mathematics, physics, computer science, and engineering majors at the undergraduate and graduate levels. It also serves as a valuable reference for further study in any discipline that requires a firm understanding of mathematical techniques and concepts.
... Singer , I . : On best approximation in vector valued norms . Collog . Math . Soc
. Ianos Bolyai 19 . Fourier Analysis and Approx . Theory , ( 1976 ) . 89 - 100 .
Bacopoulos , A . and Singer , I . : On convex vectorial optimization in linear spaces .
Author: George Isac
Publisher: Peter Lang Pub Incorporated
This book presents several new results on the best simultaneous approximation in locally convex spaces. The concept of nuclear cone is systematically used to establish some interesting relations with Pareto optimization and the duality for vectorial optimization programs with multifunctions.
Namely , following [ 1 ] , one can consider on E the " norm ” with values in the
plane R2 ( with its natural partial ordering ) , defined ... with values in partially
ordered linear spaces see e.g. [ 7 ] ) and to call an element 80 € G an element of
best vectorial approximation of x ... that VG ( x ) = g ( x ) may also happen for the vector - valued norm ( 5 ) , namely , in the particular case when llolly = 11 • 112 =
|| • 1 ) .
4 ) the Hilbert space norm | | | | is used since we are dealing with a vector valued
kernel . In order to apply transference methods to this more general situation we
can use the following result : LP ( R ) having ( 3 . 6 ) The Hilbert space valued ...
A continuation of the authors' previous book, Isometries on Banach Spaces: Vector-valued Function Spaces and Operator Spaces, Volume Two covers much of the work that has been done on characterizing isometries on various Banach spaces.
Author: Richard J. Fleming
Publisher: CRC Press
A continuation of the authors’ previous book, Isometries on Banach Spaces: Vector-valued Function Spaces and Operator Spaces, Volume Two covers much of the work that has been done on characterizing isometries on various Banach spaces. Picking up where the first volume left off, the book begins with a chapter on the Banach–Stone property. The authors consider the case where the isometry is from C0(Q, X) to C0(K, Y) so that the property involves pairs (X, Y) of spaces. The next chapter examines spaces X for which the isometries on LP(μ, X) can be described as a generalization of the form given by Lamperti in the scalar case. The book then studies isometries on direct sums of Banach and Hilbert spaces, isometries on spaces of matrices with a variety of norms, and isometries on Schatten classes. It subsequently highlights spaces on which the group of isometries is maximal or minimal. The final chapter addresses more peripheral topics, such as adjoint abelian operators and spectral isometries. Essentially self-contained, this reference explores a fundamental aspect of Banach space theory. Suitable for both experts and newcomers to the field, it offers many references to provide solid coverage of the literature on isometries.
Let E be a lienar space over the ( real or complex ) field K endomed with a ( vector - valued ) norm 1 : 1 with values in the plane R ? ( with ... ( the distance of a
point to a subset of a normed linear space ) for the subsets of the plane R ?, DIST
... ( 2006d : 41041 ) 41465 46E15 Stepanets , A . I . ( UKR - AOS ; Kiev ) Extremal
problems of approximation theory in linear spaces . ... It is shown that a simple
modification of the qd - algorithm makes it possible to construct the
homogeneous multivariate Padé ... ( 3 ) Co ( T , X ) , for a locally compact
Hausdorff space T , equipped with the strict topology defined by all weighted seminorms of the form Pw : C ...
approximation. A variety of problems involving vector spaces and linear
operators can only be addressed if we can measure the ... A norm is a real- valued function defined on a vector space that defines the size of the vectors in
Author: Mark S. Gockenbach
Publisher: CRC Press
Linear algebra forms the basis for much of modern mathematics—theoretical, applied, and computational. Finite-Dimensional Linear Algebra provides a solid foundation for the study of advanced mathematics and discusses applications of linear algebra to such diverse areas as combinatorics, differential equations, optimization, and approximation. The author begins with an overview of the essential themes of the book: linear equations, best approximation, and diagonalization. He then takes students through an axiomatic development of vector spaces, linear operators, eigenvalues, norms, and inner products. In addition to discussing the special properties of symmetric matrices, he covers the Jordan canonical form, an important theoretical tool, and the singular value decomposition, a powerful tool for computation. The final chapters present introductions to numerical linear algebra and analysis in vector spaces, including a brief introduction to functional analysis (infinite-dimensional linear algebra). Drawing on material from the author’s own course, this textbook gives students a strong theoretical understanding of linear algebra. It offers many illustrations of how linear algebra is used throughout mathematics.
As before , lfl stands for the norm of f when f e B. In case B is an inner product
space , this norm can be taken to be the one induced by the inner product . Also ,
in case B is only a normed linear space , in the proof of Theorem 4.4 the
... Volterra series, such as its convergence, approximation of a digital nonlinear
system by the discrete Volterra series, and so on. All the ... Here, for the first time
in this book, the first notion of the functional analysis is introduced, that is, of the norm of a vector, and of a vector-valued sequence. ... linear space of which
elements are vectors, or scalar- or vectorvalued sequences, normed space,
metric space, ...
Author: Andrzej Borys
Publisher: CRC Press
Category: Technology & Engineering
The discrete Volterra series holds particular value in the analysis of nonlinear systems in telecommunications. However, most books on the Volterra series either do not address this application or only offer a partial discussion. Nonlinear Aspects of Telecommunications provides an in-depth treatment of the Volterra series and the benefits it offers as a representation of nonlinear problems, particularly in echo cancellation in digital telecommunications systems. Beginning with the fundamentals of the discrete Volterra series, the author presents its basic definition, notions, conditions for convergence and stability, and its matrix representation for multiple-input and multiple-output nonlinear digital systems. He pays significant attention to the important problem of approximating a nonlinear digital system using the discrete Volterra series and offers new results in this area--results not yet available in other texts. The second part of the book uses the background of Part I to show the Volterra series' application to echo cancellation. It provides introductory material regarding the basics of adaptive cancellers, and analyzes structures for nonlinear echo cancellers using nonlinear transversal filters for baseband transmission. The last section covers nonlinear echo cancellers for voiceband transmission and interleaved structures. Full of illustrations, examples, and new results, Nonlinear Aspects of Telecommunications is your first and best resource for understanding and applying the discrete Volterra series to nonlinear echo cancellation problems. Features
Typical examples are scalar or vector valued functions defined on complex 2 or 3
dimensional meshes. Grosso et al. presented a method for mesh optimization
based on finite elements approximations with the L? norm and adaptive local
mesh refinement. ... In this paper we present a technique based on linear approximations in Hilbert spaces and the finite element method to generate
sequences of ...
Author: H.-C. Hege
Publisher: Springer Science & Business Media
Mathematical Visualization is a young new discipline. It offers efficient visualization tools to the classical subjects of mathematics, and applies mathematical techniques to problems in computer graphics and scientific visualization. Originally, it started in the interdisciplinary area of differential geometry, numerical mathematics, and computer graphics. In recent years, the methods developed have found important applications. The current volume is the quintessence of an international workshop in September 1997 in Berlin, focusing on recent developments in this emerging area. Experts present selected research work on new algorithms for visualization problems, describe the application and experiments in geometry, and develop new numerical or computer graphical techniques.
JUST NON - METABELIAN GROUPS are shown to have close relations to
simultaneous approximations . ... Harry W . McLaughlin The problem we consider
is the approximation of m elements ( x1 , . . . , xm ) in a normed linear space by
one element in a subset of that normed linear space . ... It has been shown that
more flexibility is achieved by using generalized spaces where the norm is vector valued .