Markov Operators, Positive Semigroups and Approximation Processes.

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.

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