Operator Inequalities of Ostrowski and Trapezoidal Type [electronic resource] /
Dragomir, Silvestru Sever.author.
Inequalities of Ostrowski and Trapezoidal Type for Functions of Selfadjoint Operators on Hilbert Spaces presents recent results concerning Ostrowski and Trapezoidal type inequalities for continuous functions of bounded Selfadjoint operators on complex Hilbert spaces. The first chapter recalls some fundamental facts concerning bounded Selfadjoint operators on complex Hilbert spaces. The generalized Schwarz’s inequality for positive Selfadjoint operators as well as some results for the spectrum of this class of operators are presented. The author also introduces and explores the fundamental results for polynomials in a linear operator, continuous functions of selfadjoint operators that will play a central role throughout the book. The following chapter is devoted to the Ostrowski’s type inequalities, which provide sharp error estimates in approximating the value of a function by its integral mean and can be used to obtain a priory error bounds for different quadrature rules in approximating the Riemann integral by different Riemann sums. The author also presents recent results extending Ostrowski inequality in various directions for continuous functions of selfadjoint operators in complex Hilbert spaces. The final chapter illustrates recent results obtained in extending trapezoidal type inequality in various directions for continuous functions of selfadjoint operators in complex Hilbert spaces. Applications for mid-point inequalities and some elementary functions of operators as also provided. This book is intended for use by researchers in various fields of Linear Operator Theory and Mathematical Inequalities. As well as postgraduate students and scientists applying inequalities in their specific areas.
Description based upon print version of record.
Operator Inequalities of Ostrowski and Trapezoidal Type; Preface; Contents; Chapter 1 Background for Functions of Selfadjoint Operators; 1.1 Introduction; 1.2 Bounded Selfadjoint Operators; 1.2.1 Operator Order; 1.3 Continuous Functions of Selfadjoint Operators; 1.3.1 Polynomials in a Bounded Operator; 1.3.2 Continuous Functions of Selfadjoint Operators; 1.4 Step Functions of Selfadjoint Operators; 1.5 The Spectral Representation of Selfadjoint Operators; References; Chapter 2 Inequalities of Ostrowski Type; 2.1 Introduction; 2.2 Scalar Ostrowski's Type Inequalities
2.3 Ostrowski's Type Inequalities for Hölder Continuous Functions2.3.1 Introduction; 2.3.2 More Inequalities of Ostrowski's Type; 2.3.3 The Case of ( ,) -Lipschitzian Functions; 2.3.4 Related Results; 2.4 Other Ostrowski Inequalities for Continuous Functions; 2.4.1 Inequalities for Absolutely Continuous Functions of Selfadjoint Operators; 2.4.2 Inequalities for Convex Functions of Selfadjoint Operators; 2.4.3 Some Vector Inequalities; 2.5 More Ostrowski's Type Inequalities; 2.5.1 Some Vector Inequalities for Functions of Bounded Variation
2.5.2 Some Vector Inequalities for Lipshitzian Functions2.6 Some Vector Inequalities for Monotonic Functions; 2.7 Ostrowski's Type Vector Inequalities; 2.7.1 Some Vector Inequalities; 2.8 Bounds for the Difference Between Functions and Integral Means; 2.8.1 Vector Inequalities via Ostrowski's Type Bounds; 2.8.2 Other Vector Inequalities; 2.9 Ostrowski's Type Inequalities for n-Time Differentiable Functions; 2.9.1 Some Identities; 2.9.2 Error Bounds for f( n)  of Bounded Variation; 2.9.3 Error Bounds for f( n)  Absolutely Continuous; References; Chapter 3 Inequalities of Trapezoidal Type
3.1 Introduction3.2 Scalar Trapezoidal Type Inequalities; 3.3 Trapezoidal Vector Inequalities; 3.3.1 Some General Results; 3.3.2 Other Trapezoidal Vector Inequalities; 3.4 Generalized Trapezoidal Inequalities; 3.4.1 Some Vector Inequalities; 3.5 More Generalized Trapezoidal Inequalities; 3.5.1 Other Vector Inequalities; 3.5.2 Inequalities in the Operator Order; 3.5.3 More Inequalities for Differentiable Functions; 3.6 Product Inequalities; 3.6.1 Some Vector Inequalities; References
Includes bibliographical references.
Inequalities of Ostrowski and Trapezoidal Type for Functions of Selfadjoint Operators on Hilbert Spaces presents recent results concerning Ostrowski and Trapezoidal type inequalities for continuous functions of bounded Selfadjoint operators on complex Hilbert spaces. The first chapter recalls some fundamental facts concerning bounded Selfadjoint operators on complex Hilbert spaces. The generalized Schwarz’s inequality for positive Selfadjoint operators as well as some results for the spectrum of this class of operators are presented. The author also introduces and explores the fundamental results for polynomials in a linear operator, continuous functions of selfadjoint operators that will play a central role throughout the book. The following chapter is devoted to the Ostrowski’s type inequalities, which provide sharp error estimates in approximating the value of a function by its integral mean and can be used to obtain a priory error bounds for different quadrature rules in approximating the Riemann integral by different Riemann sums. The author also presents recent results extending Ostrowski inequality in various directions for continuous functions of selfadjoint operators in complex Hilbert spaces. The final chapter illustrates recent results obtained in extending trapezoidal type inequality in various directions for continuous functions of selfadjoint operators in complex Hilbert spaces. Applications for mid-point inequalities and some elementary functions of operators as also provided. This book is intended for use by researchers in various fields of Linear Operator Theory and Mathematical Inequalities. As well as postgraduate students and scientists applying inequalities in their specific areas.
Partial differential equations.
Approximation theory.
Numerical analysis.
Mathematical optimization.
Probabilities.
Partial Differential Equations.
Approximations and Expansions.
Numerical Analysis.
Optimization.
Probability Theory and Stochastic Processes.
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