Mathematical analysis and applications : selected topics / edited by Michael Ruzhansky, Hemen Dutta, Ravi P. Agarwal.

Format:

Book

Contributor:

Ruzhansky, M. (Michael), editor.

Dutta, Hemen, 1981- editor.

Agarwal, Ravi P., editor.

Language:

English

Subjects (All):

Mathematical analysis.

Physical Description:

1 online resource (766 pages)

Edition:

1st ed.

Place of Publication:

Hoboken, New Jersey : John Wiley & Sons, Inc., [2018]

Summary:

An authoritative text that presents the current problems, theories, and applications of mathematical analysis research Mathematical Analysis and Applications: Selected Topics offers the theories, methods, and applications of a variety of targeted topics including: operator theory, approximation theory, fixed point theory, stability theory, minimization problems, many-body wave scattering problems, Basel problem, Corona problem, inequalities, generalized normed spaces, variations of functions and sequences, analytic generalizations of the Catalan, Fuss, and Fuss-Catalan Numbers, asymptotically developable functions, convex functions, Gaussian processes, image analysis, and spectral analysis and spectral synthesis. The authors-a noted team of international researchers in the field- highlight the basic developments for each topic presented and explore the most recent advances made in their area of study. The text is presented in such a way that enables the reader to follow subsequent studies in a burgeoning field of research. This important text: * Presents a wide-range of important topics having current research importance and interdisciplinary applications such as game theory, image processing, creation of materials with a desired refraction coefficient, etc. * Contains chapters written by a group of esteemed researchers in mathematical analysis Includes problems and research questions in order to enhance understanding of the information provided * Offers references that help readers advance to further study Written for researchers, graduate students, educators, and practitioners with an interest in mathematical analysis, Mathematical Analysis and Applications: Selected Topics includes the most recent research from a range of mathematical fields.

Contents:

Cover

Title Page

Copyright

Contents

Preface

About the Editors

List of Contributors

Chapter 1 Spaces of Asymptotically Developable Functions and Applications

1.1 Introduction and Some Notations

1.2 Strong Asymptotic Expansions

1.3 Monomial Asymptotic Expansions

1.4 Monomial Summability for Singularly Perturbed Differential Equations

1.5 Pfaffian Systems

References

Chapter 2 Duality for Gaussian Processes from Random Signed Measures

2.1 Introduction

2.2 Reproducing Kernel Hilbert Spaces (RKHSs) in the Measurable Category

2.3 Applications to Gaussian Processes

2.4 Choice of Probability Space

2.5 A Duality

2.A Stochastic Processes

2.B Overview of Applications of RKHSs

Acknowledgments

Chapter 3 Many‐Body Wave Scattering Problems for Small Scatterers and Creating Materials with a Desired Refraction Coefficient

3.1 Introduction

3.2 Derivation of the Formulas for One‐Body Wave Scattering Problems

3.3 Many‐Body Scattering Problem

3.3.1 The Case of Acoustically Soft Particles

3.3.2 Wave Scattering by Many Impedance Particles

3.4 Creating Materials with a Desired Refraction Coefficient

3.5 Scattering by Small Particles Embedded in an Inhomogeneous Medium

3.6 Conclusions

Chapter 4 Generalized Convex Functions and their Applications

4.1 Brief Introduction

4.2 Generalized E‐Convex Functions

4.3 Ea- Epigraph

4.4 Generalized s‐Convex Functions

4.5 Applications to Special Means

Chapter 5 Some Properties and Generalizations of the Catalan, Fuss, and Fuss-Catalan Numbers

5.1 The Catalan Numbers

5.1.1 A Definition of the Catalan Numbers

5.1.2 The History of the Catalan Numbers

5.1.3 A Generating Function of the Catalan Numbers

5.1.4 Some Expressions of the Catalan Numbers.

5.1.5 Integral Representations of the Catalan Numbers

5.1.6 Asymptotic Expansions of the Catalan Function

5.1.7 Complete Monotonicity of the Catalan Numbers

5.1.8 Inequalities of the Catalan Numbers and Function

5.1.9 The Bell Polynomials of the Second Kind and the Bessel Polynomials

5.2 The Catalan-Qi Function

5.2.1 The Fuss Numbers

5.2.2 A Definition of the Catalan-Qi Function

5.2.3 Some Identities of the Catalan-Qi Function

5.2.4 Integral Representations of the Catalan-Qi Function

5.2.5 Asymptotic Expansions of the Catalan-Qi Function

5.2.6 Complete Monotonicity of the Catalan-Qi Function

5.2.7 Schur‐Convexity of the Catalan-Qi Function

5.2.8 Generating Functions of the Catalan-Qi Numbers

5.2.9 A Double Inequality of the Catalan-Qi Function

5.2.10 The q‐Catalan-Qi Numbers and Properties

5.2.11 The Catalan Numbers and the k‐Gamma and k‐Beta Functions

5.2.12 Series Identities Involving the Catalan Numbers

5.3 The Fuss-Catalan Numbers

5.3.1 A Definition of the Fuss-Catalan Numbers

5.3.2 A Product‐Ratio Expression of the Fuss-Catalan Numbers

5.3.3 Complete Monotonicity of the Fuss-Catalan Numbers

5.3.4 A Double Inequality for the Fuss-Catalan Numbers

5.4 The Fuss-Catalan-Qi Function

5.4.1 A Definition of the Fuss-Catalan-Qi Function

5.4.2 A Product‐Ratio Expression of the Fuss-Catalan-Qi Function

5.4.3 Integral Representations of the Fuss-Catalan-Qi Function

5.4.4 Complete Monotonicity of the Fuss-Catalan-Qi Function

5.5 Some Properties for Ratios of Two Gamma Functions

5.5.1 An Integral Representation and Complete Monotonicity

5.5.2 An Exponential Expansion for the Ratio of Two Gamma Functions

5.5.3 A Double Inequality for the Ratio of Two Gamma Functions

5.6 Some New Results on the Catalan Numbers

5.7 Open Problems

References.

Chapter 6 Trace Inequalities of Jensen Type for Self‐adjoint Operators in Hilbert Spaces: A Survey of Recent Results

6.1 Introduction

6.1.1 Jensen's Inequality

6.1.2 Traces for Operators in Hilbert Spaces

6.2 Jensen's Type Trace Inequalities

6.2.1 Some Trace Inequalities for Convex Functions

6.2.2 Some Functional Properties

6.2.3 Some Examples

6.2.4 More Inequalities for Convex Functions

6.3 Reverses of Jensen's Trace Inequality

6.3.1 A Reverse of Jensen's Inequality

6.3.2 Some Examples

6.3.3 Further Reverse Inequalities for Convex Functions

6.3.4 Some Examples

6.3.5 Reverses of Hölder's Inequality

6.4 Slater's Type Trace Inequalities

6.4.1 Slater's Type Inequalities

6.4.2 Further Reverses

Chapter 7 Spectral Synthesis and Its Applications

7.1 Introduction

7.2 Basic Concepts and Function Classes

7.3 Discrete Spectral Synthesis

7.4 Nondiscrete Spectral Synthesis

7.5 Spherical Spectral Synthesis

7.6 Spectral Synthesis on Hypergroups

7.7 Applications

Chapter 8 Various Ulam-Hyers Stabilities of Euler-Lagrange-Jensen General (a,b

k=a+b)‐Sextic Functional Equations

8.1 Brief Introduction

8.2 General Solution of Euler-Lagrange-Jensen General (a,b

k=a+b)‐Sextic Functional Equation

8.3 Stability Results in Banach Space

8.3.1 Banach Space: Direct Method

8.3.2 Banach Space: Fixed Point Method

8.4 Stability Results in Felbin's Type Spaces

8.4.1 Felbin's Type Spaces: Direct Method

8.4.2 Felbin's Type Spaces: Fixed Point Method

8.5 Intuitionistic Fuzzy Normed Space: Stability Results

8.5.1 IFNS: Direct Method

8.5.2 IFNS: Fixed Point Method

Chapter 9 A Note on the Split Common Fixed Point Problem and its Variant Forms

9.1 Introduction

9.2 Basic Concepts and Definitions.

9.2.1 Introduction

9.2.2 Vector Space

9.2.3 Hilbert Space and its Properties

9.2.4 Bounded Linear Map and its Properties

9.2.5 Some Nonlinear Operators

9.2.6 Problem Formulation

9.2.7 Preliminary Results

9.2.8 Strong Convergence for the Split Common Fixed‐Point Problems for Total Quasi‐Asymptotically Nonexpansive Mappings

9.2.9 Strong Convergence for the Split Common Fixed‐Point Problems for Demicontractive Mappings

9.2.10 Application to Variational Inequality Problems

9.2.11 On Synchronal Algorithms for Fixed and Variational Inequality Problems in Hilbert Spaces

9.2.12 Preliminaries

9.3 A Note on the Split Equality Fixed‐Point Problems in Hilbert Spaces

9.3.1 Problem Formulation

9.3.2 Preliminaries

9.3.3 The Split Feasibility and Fixed‐Point Equality Problems for Quasi‐Nonexpansive Mappings in Hilbert Spaces

9.3.4 The Split Common Fixed‐Point Equality Problems for Quasi‐Nonexpansive Mappings in Hilbert Spaces

9.4 Numerical Example

9.5 The Split Feasibility and Fixed Point Problems for Quasi‐Nonexpansive Mappings in Hilbert Spaces

9.5.1 Problem Formulation

9.5.2 Preliminary Results

9.6 Ishikawa‐Type Extra‐Gradient Iterative Methods for Quasi‐Nonexpansive Mappings in Hilbert Spaces

9.6.1 Application to Split Feasibility Problems

9.7 Conclusion

Chapter 10 Stabilities and Instabilities of Rational Functional Equations and Euler-Lagrange-Jensen (a,b)‐Sextic Functional Equations

10.1 Introduction

10.1.1 Growth of Functional Equations

10.1.2 Importance of Functional Equations

10.1.3 Functional Equations Relevant to Other Fields

10.1.4 Definition of Functional Equation with Examples

10.2 Ulam Stability Problem for Functional Equation

10.2.1 ϵ‐Stability of Functional Equation

10.2.2 Stability Involving Sum of Powers of Norms.

10.2.3 Stability Involving Product of Powers of Norms

10.2.4 Stability Involving a General Control Function

10.2.5 Stability Involving Mixed Product-Sum of Powers of Norms

10.2.6 Application of Ulam Stability Theory

10.3 Various Forms of Functional Equations

10.4 Preliminaries

10.5 Rational Functional Equations

10.5.1 Reciprocal Type Functional Equation

10.5.2 Solution of Reciprocal Type Functional Equation

10.5.3 Generalized Hyers-Ulam Stability of Reciprocal Type Functional Equation

10.5.4 Counter‐Example

10.5.5 Geometrical Interpretation of Reciprocal Type Functional Equation

10.5.6 An Application of Equation (10.41) to Electric Circuits

10.5.7 Reciprocal‐Quadratic Functional Equation

10.5.8 General Solution of Reciprocal‐Quadratic Functional Equation

10.5.9 Generalized Hyers-Ulam Stability of Reciprocal‐Quadratic Functional Equations

10.5.10 Counter‐Examples

10.5.11 Reciprocal‐Cubic and Reciprocal‐Quartic Functional Equations

10.5.12 Hyers-Ulam Stability of Reciprocal‐Cubic and Reciprocal‐Quartic Functional Equations

10.5.13 Counter‐Examples

10.6 Euler‐Lagrange-Jensen (a,b

10.6.1 Generalized Ulam-Hyers Stability of Euler‐Lagrange‐Jensen Sextic Functional Equation Using Fixed Point Method

10.6.2 Counter‐Example

10.6.3 Generalized Ulam-Hyers Stability of Euler‐Lagrange‐Jensen Sextic Functional Equation Using Direct Method

Chapter 11 Attractor of the Generalized Contractive Iterated Function System

11.1 Iterated Function System

11.2 Generalized F‐contractive Iterated Function System

11.3 Iterated Function System in b‐Metric Space

11.4 Generalized F‐Contractive Iterated Function System in b‐Metric Space

Chapter 12 Regular and Rapid Variations and Some Applications.

12.1 Introduction and Historical Background.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9781119414339

1119414334

9781119414308

111941430X

9781119414421

1119414423

OCLC:

1031967446