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Author Altomare, Francesco
Title Markov Operators, Positive Semigroups and Approximation Processes
Imprint Berlin/Boston : De Gruyter, Inc., 2014
book jacket
Descript 1 online resource (326 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Series De Gruyter Studies in Mathematics Ser. ; v.61
De Gruyter Studies in Mathematics Ser
Note Intro -- Preface -- Introduction -- Guide to the reader and interdependence of sections -- Notation -- 1 Positive linear operators and approximation problems -- 1.1 Positive linear functionals and operators -- 1.1.1 Positive Radon measures -- 1.1.2 Choquet boundaries -- 1.1.3 Bauer simplices -- 1.2 Korovkin-type approximation theorems -- 1.3 Further convergence criteria for nets of positive linear operators -- 1.4 Asymptotic behaviour of Lipschitz contracting Markov semigroups -- 1.5 Asymptotic formulae for positive linear operators -- 1.6 Moduli of smoothness and degree of approximation by positive linear operators -- 1.7 Notes and comments -- 2 C0-semigroups of operators and linear evolution equations -- 2.1 C0-semigroups of operators and abstract Cauchy problems -- 2.1.1 C0-semigroups and their generators -- 2.1.2 Generation theorems and abstract Cauchy problems -- 2.2 Approximation of C0-semigroups -- 2.3 Feller and Markov semigroups of operators -- 2.3.1 Basic properties -- 2.3.2 Markov Processes -- 2.3.3 Second-order differential operators on real intervals and Feller theory -- 2.3.4 Multidimensional second-order differential operators and Markov semigroups -- 2.4 Notes and comments -- 3 Bernstein-Schnabl operators associated with Markov operators -- 3.1 Generalities, definitions and examples -- 3.1.1 Bernstein-Schnabl operators on [0,1] -- 3.1.2 Bernstein-Schnabl operators on Bauer simplices -- 3.1.3 Bernstein operators on polytopes -- 3.1.4 Bernstein-Schnabl operators associated with strictly elliptic differential operators -- 3.1.5 Bernstein-Schnabl operators associated with tensor products of Markov operators -- 3.1.6 Bernstein-Schnabl operators associated with convex combinations of Markov operators -- 3.1.7 Bernstein-Schnabl operators associated with convex convolution products of Markov operators
3.2 Approximation properties and rate of convergence -- 3.3 Preservation of Hölder continuity -- 3.3.1 Smallest Lipschitz constants and triangles -- 3.3.2 Smallest Lipschitz constants and parallelograms -- 3.4 Bernstein-Schnabl operators and convexity -- 3.5 Monotonicity properties -- 3.6 Notes and comments -- 4 Differential operators and Markov semigroups associated with Markov operators -- 4.1 Asymptotic formulae for Bernstein-Schnabl operators -- 4.2 Differential operators associated with Markov operators -- 4.3 Markov semigroups generated by differential operators associated with Markov operators -- 4.4 Preservation properties and asymptotic behaviour -- 4.5 The special case of the unit interval -- 4.5.1 Degenerate differential operators on [0,1] -- 4.5.2 Approximation properties by means of Bernstein-Schnabl operators -- 4.5.3 Preservation properties and asymptotic behaviour -- 4.5.4 The saturation class of Bernstein-Schnabl operators and the Favard class of their limit semigroups -- 4.6 Notes and comments -- 5 Perturbed differential operators and modified Bernstein-Schnabl operators -- 5.1 Lototsky-Schnabl operators -- 5.2 A modification of Bernstein-Schnabl operators -- 5.3 Approximation properties -- 5.4 Preservation properties -- 5.5 Asymptotic formulae -- 5.6 Modified Bernstein-Schnabl operators and first-order perturbations -- 5.7 The unit interval -- 5.7.1 Complete degenerate second-order differential operators on [0, 1] -- 5.7.2 Approximation properties by means of modified Bernstein-Schnabl operators -- 5.8 The d-dimensional simplex and hypercube -- 5.9 Notes and comments -- Appendices -- A.1 A classification of Markov operators on two dimensional convex compact subsets -- A.2 Rate of convergence for the limit semigroup of Bernstein operators -- Bibliography -- Symbol index -- Index -- Leere Seite
The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob
Description based on publisher supplied metadata and other sources
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Link Print version: Altomare, Francesco Markov Operators, Positive Semigroups and Approximation Processes Berlin/Boston : De Gruyter, Inc.,c2014 9783110372748
Subject Boundary value problems.;Differential operators.;Semigroups.;Markov operators
Electronic books
Alt Author Cappelletti, Mirella
Leonessa, Vita
Rasa, Ioan
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