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After an introduction to the geometry of polynomials and a discussion of refinements of the Fundamental Theorem of Algebra, the book turns to a consideration of various special polynomials. Chebyshev and Descartes systems are then introduced, and Müntz systems and rational systems are examined in detail. Subsequent chapters discuss denseness questions and the inequalities satisfied by polynomials and rational functions. Appendices on algorithms and computational concerns, on the interpolation theorem, and on orthogonality and irrationality round off the text. The book is self-contained and assumes at most a senior-undergraduate familiarity with real and complex analysis.
Book Synopsis Polynomials and Polynomial Inequalities by : Peter Borwein
Download or read book Polynomials and Polynomial Inequalities written by Peter Borwein and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 491 pages. Available in PDF, EPUB and Kindle. Book excerpt: After an introduction to the geometry of polynomials and a discussion of refinements of the Fundamental Theorem of Algebra, the book turns to a consideration of various special polynomials. Chebyshev and Descartes systems are then introduced, and Müntz systems and rational systems are examined in detail. Subsequent chapters discuss denseness questions and the inequalities satisfied by polynomials and rational functions. Appendices on algorithms and computational concerns, on the interpolation theorem, and on orthogonality and irrationality round off the text. The book is self-contained and assumes at most a senior-undergraduate familiarity with real and complex analysis.
The book contains some of the most important results on the analysis of polynomials and their derivatives. Besides the fundamental results which are treated with their proofs, the book also provides an account of the most recent developments concerning extremal properties of polynomials and their derivatives in various metrics with an extensive analysis of inequalities for trigonometric sums and algebraic polynomials, as well as their zeros. The final chapter provides some selected applications of polynomials in approximation theory and computer aided geometric design (CAGD). One can also find in this book several new research problems and conjectures with sufficient information concerning the results obtained to date towards the investigation of their solution. Contents:PrefaceGeneral Concept of Algebraic PolynomialsSelected Polynomial InequalitiesZeros of PolynomialsInequalities Connected with Trigonometric SumsExtremal Problems for PolynomialsExtremal Problems of Markov-Bernstein TypeSome Applications of PolynomialsSymbol IndexName IndexSubject Index Readership: Mathematicians and mathematical physicists. keywords:Algebraic Polynomials;Trigonometric Polynomials;Zeros;Extremal Problems;Trigonometric Sums;Positivity and Monotonicity;Distribution of Zeros;Bounds for Polynomial Zeros;Incomplete Polynomials;Polynomials with Minimal Norm;Markov-Bernstein Inequalities;Approximation;Symmetric Functions;Orthogonal Polynomials;Nonnegative Polynomials “The topics are tastefully selected and the results are easy to find. Although this book is not really planned as a textbook to teach from, it is excellent for self-study or seminars. This is a very useful reference book with many results which have not appeared in a book form yet. It is an important addition to the literature.” Journal of Approximation Theory “I find the book to be well written and readable. The authors have made an attempt to present the material in an integrated and self-contained fashion and, in my opinion, they have been greatly successful. The book would be useful not only for the specialist mathematician, but also for those researchers in the applied and computational sciences who use polynomials as a tool.” Mathematical Reviews “This is a remarkable book, offering a cornucopia of results, all connected by their involvement with polynomials. The scope of the volume can be conveyed by citing some statistics: there are 821 pages, 7 chapters, 20 sections, 108 subsections, 95 pages of references (distributed throughout the book), a name index of 16 pages, and a subject index of 19 pages … The book is written in a gentle style: one can open it anywhere and begin to understand, without encountering unfamiliar notation and terminology. It is strongly recommended to individuals and to libraries.” Mathematics of Computation “This book contains some of the most important results on the analysis of polynomials and their derivatives … is intended, not only for the specialist mathematician, but also for those researchers in the applied sciences who use polynomials as a tool.” Sever S Dragomir “This is a well-written book on a widely useful topic. It is strongly recommended not only to the mathematical specialist, but also to all those researchers in the applied and computational sciences who make frequent use of polynomials as a tool. Of course, libraries will also benefit greatly by including this book in their cherished collection.” Mathematics Abstracts “There is no doubt that this is a very useful work compiling enormous researches carried out on the subject … This is a well-written book on a widely useful topic.” Zentralblatt für Mathematik
Book Synopsis Topics in Polynomials by : G V Milovanovic
Download or read book Topics in Polynomials written by G V Milovanovic and published by World Scientific. This book was released on 1994-06-28 with total page 836 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book contains some of the most important results on the analysis of polynomials and their derivatives. Besides the fundamental results which are treated with their proofs, the book also provides an account of the most recent developments concerning extremal properties of polynomials and their derivatives in various metrics with an extensive analysis of inequalities for trigonometric sums and algebraic polynomials, as well as their zeros. The final chapter provides some selected applications of polynomials in approximation theory and computer aided geometric design (CAGD). One can also find in this book several new research problems and conjectures with sufficient information concerning the results obtained to date towards the investigation of their solution. Contents:PrefaceGeneral Concept of Algebraic PolynomialsSelected Polynomial InequalitiesZeros of PolynomialsInequalities Connected with Trigonometric SumsExtremal Problems for PolynomialsExtremal Problems of Markov-Bernstein TypeSome Applications of PolynomialsSymbol IndexName IndexSubject Index Readership: Mathematicians and mathematical physicists. keywords:Algebraic Polynomials;Trigonometric Polynomials;Zeros;Extremal Problems;Trigonometric Sums;Positivity and Monotonicity;Distribution of Zeros;Bounds for Polynomial Zeros;Incomplete Polynomials;Polynomials with Minimal Norm;Markov-Bernstein Inequalities;Approximation;Symmetric Functions;Orthogonal Polynomials;Nonnegative Polynomials “The topics are tastefully selected and the results are easy to find. Although this book is not really planned as a textbook to teach from, it is excellent for self-study or seminars. This is a very useful reference book with many results which have not appeared in a book form yet. It is an important addition to the literature.” Journal of Approximation Theory “I find the book to be well written and readable. The authors have made an attempt to present the material in an integrated and self-contained fashion and, in my opinion, they have been greatly successful. The book would be useful not only for the specialist mathematician, but also for those researchers in the applied and computational sciences who use polynomials as a tool.” Mathematical Reviews “This is a remarkable book, offering a cornucopia of results, all connected by their involvement with polynomials. The scope of the volume can be conveyed by citing some statistics: there are 821 pages, 7 chapters, 20 sections, 108 subsections, 95 pages of references (distributed throughout the book), a name index of 16 pages, and a subject index of 19 pages … The book is written in a gentle style: one can open it anywhere and begin to understand, without encountering unfamiliar notation and terminology. It is strongly recommended to individuals and to libraries.” Mathematics of Computation “This book contains some of the most important results on the analysis of polynomials and their derivatives … is intended, not only for the specialist mathematician, but also for those researchers in the applied sciences who use polynomials as a tool.” Sever S Dragomir “This is a well-written book on a widely useful topic. It is strongly recommended not only to the mathematical specialist, but also to all those researchers in the applied and computational sciences who make frequent use of polynomials as a tool. Of course, libraries will also benefit greatly by including this book in their cherished collection.” Mathematics Abstracts “There is no doubt that this is a very useful work compiling enormous researches carried out on the subject … This is a well-written book on a widely useful topic.” Zentralblatt für Mathematik
This volume, which presents the cumulation of the authors' research in the field, deals with Lidstone, Hermite, Abel--Gontscharoff, Birkhoff, piecewise Hermite and Lidstone, spline and Lidstone--spline interpolating problems. Explicit representations of the interpolating polynomials and associated error functions are given, as well as explicit error inequalities in various norms. Numerical illustrations are provided of the importance and sharpness of the various results obtained. Also demonstrated are the significance of these results in the theory of ordinary differential equations such as maximum principles, boundary value problems, oscillation theory, disconjugacy and disfocality. For mathematicians, numerical analysts, computer scientists and engineers.
Book Synopsis Error Inequalities in Polynomial Interpolation and Their Applications by : R.P. Agarwal
Download or read book Error Inequalities in Polynomial Interpolation and Their Applications written by R.P. Agarwal and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 376 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume, which presents the cumulation of the authors' research in the field, deals with Lidstone, Hermite, Abel--Gontscharoff, Birkhoff, piecewise Hermite and Lidstone, spline and Lidstone--spline interpolating problems. Explicit representations of the interpolating polynomials and associated error functions are given, as well as explicit error inequalities in various norms. Numerical illustrations are provided of the importance and sharpness of the various results obtained. Also demonstrated are the significance of these results in the theory of ordinary differential equations such as maximum principles, boundary value problems, oscillation theory, disconjugacy and disfocality. For mathematicians, numerical analysts, computer scientists and engineers.
The study of positive polynomials brings together algebra, geometry and analysis. The subject is of fundamental importance in real algebraic geometry when studying the properties of objects defined by polynomial inequalities. Hilbert's 17th problem and its solution in the first half of the 20th century were landmarks in the early days of the subject. More recently, new connections to the moment problem and to polynomial optimization have been discovered. The moment problem relates linear maps on the multidimensional polynomial ring to positive Borel measures. This book provides an elementary introduction to positive polynomials and sums of squares, the relationship to the moment problem, and the application to polynomial optimization. The focus is on the exciting new developments that have taken place in the last 15 years, arising out of Schmudgen's solution to the moment problem in the compact case in 1991. The book is accessible to a well-motivated student at the beginning graduate level. The objects being dealt with are concrete and down-to-earth, namely polynomials in $n$ variables with real coefficients, and many examples are included. Proofs are presented as clearly and as simply as possible. Various new, simpler proofs appear in the book for the first time. Abstraction is employed only when it serves a useful purpose, but, at the same time, enough abstraction is included to allow the reader easy access to the literature. The book should be essential reading for any beginning student in the area.
Book Synopsis Positive Polynomials and Sums of Squares by : Murray Marshall
Download or read book Positive Polynomials and Sums of Squares written by Murray Marshall and published by American Mathematical Soc.. This book was released on 2008 with total page 201 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of positive polynomials brings together algebra, geometry and analysis. The subject is of fundamental importance in real algebraic geometry when studying the properties of objects defined by polynomial inequalities. Hilbert's 17th problem and its solution in the first half of the 20th century were landmarks in the early days of the subject. More recently, new connections to the moment problem and to polynomial optimization have been discovered. The moment problem relates linear maps on the multidimensional polynomial ring to positive Borel measures. This book provides an elementary introduction to positive polynomials and sums of squares, the relationship to the moment problem, and the application to polynomial optimization. The focus is on the exciting new developments that have taken place in the last 15 years, arising out of Schmudgen's solution to the moment problem in the compact case in 1991. The book is accessible to a well-motivated student at the beginning graduate level. The objects being dealt with are concrete and down-to-earth, namely polynomials in $n$ variables with real coefficients, and many examples are included. Proofs are presented as clearly and as simply as possible. Various new, simpler proofs appear in the book for the first time. Abstraction is employed only when it serves a useful purpose, but, at the same time, enough abstraction is included to allow the reader easy access to the literature. The book should be essential reading for any beginning student in the area.
Inequalities for polynomials and their derivatives are very important in many areas of mathematics, as well as in other computational and applied sciences; in particular they play a fundamental role in approximation theory. Here, not only Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials, but also ones for trigonometric polynomials and related functions, are treated in an integrated and comprehensive style in different metrics, both on general classes of polynomials and on important restrictive classes of polynomials. Primarily for graduate and PhD students, this book is useful for any researchers exploring problems which require derivative estimates. It is particularly useful for those studying inverse problems in approximation theory. Applies Markov-Bernstein-type inequalities to any problem where derivative estimates are necessary Presents complex math in a clean and simple way, progressing readers from polynomials into rational functions, and entire functions of exponential type Contains exhaustive references with more than five hundred citations to articles and books Features methods to solve inverse problems across approximation theory Includes open problems for further research
Book Synopsis Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials by : Robert B. Gardner
Download or read book Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials written by Robert B. Gardner and published by Academic Press. This book was released on 2022-02-10 with total page 444 pages. Available in PDF, EPUB and Kindle. Book excerpt: Inequalities for polynomials and their derivatives are very important in many areas of mathematics, as well as in other computational and applied sciences; in particular they play a fundamental role in approximation theory. Here, not only Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials, but also ones for trigonometric polynomials and related functions, are treated in an integrated and comprehensive style in different metrics, both on general classes of polynomials and on important restrictive classes of polynomials. Primarily for graduate and PhD students, this book is useful for any researchers exploring problems which require derivative estimates. It is particularly useful for those studying inverse problems in approximation theory. Applies Markov-Bernstein-type inequalities to any problem where derivative estimates are necessary Presents complex math in a clean and simple way, progressing readers from polynomials into rational functions, and entire functions of exponential type Contains exhaustive references with more than five hundred citations to articles and books Features methods to solve inverse problems across approximation theory Includes open problems for further research
Book Synopsis Topics in Polynomials by : Gradimir V. Milovanovic
Download or read book Topics in Polynomials written by Gradimir V. Milovanovic and published by . This book was released on 1994 with total page 821 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Contributions by leading experts in the field provide a snapshot of current progress in polynomials and number theory.
Book Synopsis Number Theory and Polynomials by : James Fraser McKee
Download or read book Number Theory and Polynomials written by James Fraser McKee and published by Cambridge University Press. This book was released on 2008-05-08 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contributions by leading experts in the field provide a snapshot of current progress in polynomials and number theory.
Book Synopsis Intermediate Algebra 2e by : Lynn Marecek
Download or read book Intermediate Algebra 2e written by Lynn Marecek and published by . This book was released on 2020-05-06 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Positivity is one of the most basic mathematical concepts, involved in many areas of mathematics (analysis, real algebraic geometry, functional analysis, etc.). The main objective of the book is to give useful characterizations of polynomials. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed.
Book Synopsis Positive Polynomials by : Alexander Prestel
Download or read book Positive Polynomials written by Alexander Prestel and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 269 pages. Available in PDF, EPUB and Kindle. Book excerpt: Positivity is one of the most basic mathematical concepts, involved in many areas of mathematics (analysis, real algebraic geometry, functional analysis, etc.). The main objective of the book is to give useful characterizations of polynomials. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed.
This book provides a general introduction to modern mathematical aspects in computing with multivariate polynomials and in solving algebraic systems. It presents the state of the art in several symbolic, numeric, and symbolic-numeric techniques, including effective and algorithmic methods in algebraic geometry and computational algebra, complexity issues, and applications ranging from statistics and geometric modelling to robotics and vision. Graduate students, as well as researchers in related areas, will find an excellent introduction to currently interesting topics. These cover Groebner and border bases, multivariate resultants, residues, primary decomposition, multivariate polynomial factorization, homotopy continuation, complexity issues, and their applications.
Book Synopsis Solving Polynomial Equations by : Alicia Dickenstein
Download or read book Solving Polynomial Equations written by Alicia Dickenstein and published by Springer Science & Business Media. This book was released on 2005-04-27 with total page 433 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a general introduction to modern mathematical aspects in computing with multivariate polynomials and in solving algebraic systems. It presents the state of the art in several symbolic, numeric, and symbolic-numeric techniques, including effective and algorithmic methods in algebraic geometry and computational algebra, complexity issues, and applications ranging from statistics and geometric modelling to robotics and vision. Graduate students, as well as researchers in related areas, will find an excellent introduction to currently interesting topics. These cover Groebner and border bases, multivariate resultants, residues, primary decomposition, multivariate polynomial factorization, homotopy continuation, complexity issues, and their applications.
