- Preface xv About the Editors xxi List of Contributors xxiii 1 Spaces of Asymptotically Developable Functions and Applications 1Sergio Alejandro Carrillo Torres and Jorge Mozo Fernández 1.1 Introduction and Some Notations 1 1.2 Strong Asymptotic Expansions 2 1.3 Monomial Asymptotic Expansions 7 1.4 Monomial Summability for Singularly Perturbed Differential Equations 13 1.5 Pfaffian Systems 15 References 19 2 Duality for Gaussian Processes from Random Signed Measures 23Palle E.T. Jorgensen and Feng Tian 2.1 Introduction 23 2.2 Reproducing Kernel Hilbert Spaces (RKHSs) in the Measurable Category 24 2.3 Applications to Gaussian Processes 30 2.4 Choice of Probability Space 34 2.5 A Duality 37 2.A Stochastic Processes 40 2.B Overview of Applications of RKHSs 45 Acknowledgments 50 References 51 3 Many-BodyWave Scattering Problems for Small Scatterers and CreatingMaterials with a Desired Refraction Coefficient 57Alexander G. Ramm 3.1 Introduction 57 3.2 Derivation of the Formulas for One-BodyWave Scattering Problems 62 3.3 Many-Body Scattering Problem 65 3.3.1 The Case of Acoustically Soft Particles 68 3.3.2 Wave Scattering by Many Impedance Particles 70 3.4 Creating Materials with a Desired Refraction Coefficient 71 3.5 Scattering by Small Particles Embedded in an Inhomogeneous Medium 72 3.6 Conclusions 72 References 73 4 Generalized Convex Functions and their Applications 77Adem Kiliçman andWedad Saleh 4.1 Brief Introduction 77 4.2 Generalized E-Convex Functions 78 4.3 E££-Epigraph 84 4.4 Generalized s-Convex Functions 85 4.5 Applications to Special Means 96 References 98 5 Some Properties and Generalizations of the Catalan, Fuss, and FussCatalan Numbers 101Feng Qi and Bai-Ni Guo 5.1 The Catalan Numbers 101 5.1.1 A Definition of the Catalan Numbers 101 5.1.2 The History of the Catalan Numbers 101 5.1.3 A Generating Function of the Catalan Numbers 102 5.1.4 Some Expressions of the Catalan Numbers 102 5.1.5 Integral Representations of the Catalan Numbers 103 5.1.6 Asymptotic Expansions of the Catalan Function 104 5.1.7 Complete Monotonicity of the Catalan Numbers 105 5.1.8 Inequalities of the Catalan Numbers and Function 106 5.1.9 The Bell Polynomials of the Second Kind and the Bessel Polynomials 109 5.2 The CatalanQi Function 111 5.2.1 The Fuss Numbers 111 5.2.2 A Definition of the CatalanQi Function 111 5.2.3 Some Identities of the CatalanQi Function 112 5.2.4 Integral Representations of the CatalanQi Function 114 5.2.5 Asymptotic Expansions of the CatalanQi Function 115 5.2.6 Complete Monotonicity of the CatalanQi Function 116 5.2.7 Schur-Convexity of the CatalanQi Function 118 5.2.8 Generating Functions of the CatalanQi Numbers 118 5.2.9 A Double Inequality of the CatalanQi Function 118 5.2.10 The q-CatalanQi Numbers and Properties 119 5.2.11 The Catalan Numbers and the k-Gamma and k-Beta Functions 119 5.2.12 Series Identities Involving the Catalan Numbers 119 5.3 The FussCatalan Numbers 119 5.3.1 A Definition of the FussCatalan Numbers 119 5.3.2 A Product-Ratio Expression of the FussCatalan Numbers 120 5.3.3 Complete Monotonicity of the FussCatalan Numbers 120 5.3.4 A Double Inequality for the FussCatalan Numbers 121 5.4 The FussCatalanQi Function 121 5.4.1 A Definition of the FussCatalanQi Function 121 5.4.2 A Product-Ratio Expression of the FussCatalanQi Function 122 5.4.3 Integral Representations of the FussCatalanQi Function 123 5.4.4 Complete Monotonicity of the FussCatalanQi Function 124 5.5 Some Properties for Ratios of Two Gamma Functions 124 5.5.1 An Integral Representation and Complete Monotonicity 125 5.5.2 An Exponential Expansion for the Ratio of Two Gamma Functions 125 5.5.3 A Double Inequality for the Ratio of Two Gamma Functions 125 5.6 Some NewResults on the Catalan Numbers 126 5.7 Open Problems 126 Acknowledgments 127 References 127 6 Trace Inequalities of Jensen Type for Self-adjoint Operators in Hilbert Spaces: A Survey of Recent Results 135Silvestru Sever Dragomir 6.1 Introduction 135 6.1.1 Jensen's Inequality 135 6.1.2 Traces for Operators in Hilbert Spaces 138 6.2 Jensen's Type Trace Inequalities 141 6.2.1 Some Trace Inequalities for Convex Functions 141 6.2.2 Some Functional Properties 145 6.2.3 Some Examples 151 6.2.4 More Inequalities for Convex Functions 154 6.3 Reverses of Jensen's Trace Inequality 157 6.3.1 A Reverse of Jensen's Inequality 157 6.3.2 Some Examples 163 6.3.3 Further Reverse Inequalities for Convex Functions 165 6.3.4 Some Examples 169 6.3.5 Reverses of Hölder's Inequality 174 6.4 Slater's Type Trace Inequalities 177 6.4.1 Slater's Type Inequalities 177 6.4.2 Further Reverses 180 References 188 7 Spectral Synthesis and Its Applications 193László Székelyhidi 7.1 Introduction 193 7.2 Basic Concepts and Function Classes 195 7.3 Discrete Spectral Synthesis 203 7.4 Nondiscrete Spectral Synthesis 217 7.5 Spherical Spectral Synthesis 219 7.6 Spectral Synthesis on Hypergroups 238 7.7 Applications 248 Acknowledgments 252 References 252 8 Various UlamHyers Stabilities of EulerLagrangeJensen General (a, b; k = <

- Preface xv About the Editors xxi List of Contributors xxiii 1 Spaces of Asymptotically Developable Functions and Applications 1Sergio Alejandro Carrillo Torres and Jorge Mozo Fernández 1.1 Introduction and Some Notations 1 1.2 Strong Asymptotic Expansions 2 1.3 Monomial Asymptotic Expansions 7 1.4 Monomial Summability for Singularly Perturbed Differential Equations 13 1.5 Pfaffian Systems 15 References 19 2 Duality for Gaussian Processes from Random Signed Measures 23Palle E.T. Jorgensen and Feng Tian 2.1 Introduction 23 2.2 Reproducing Kernel Hilbert Spaces (RKHSs) in the Measurable Category 24 2.3 Applications to Gaussian Processes 30 2.4 Choice of Probability Space 34 2.5 A Duality 37 2.A Stochastic Processes 40 2.B Overview of Applications of RKHSs 45 Acknowledgments 50 References 51 3 Many-BodyWave Scattering Problems for Small Scatterers and CreatingMaterials with a Desired Refraction Coefficient 57Alexander G. Ramm 3.1 Introduction 57 3.2 Derivation of the Formulas for One-BodyWave Scattering Problems 62 3.3 Many-Body Scattering Problem 65 3.3.1 The Case of Acoustically Soft Particles 68 3.3.2 Wave Scattering by Many Impedance Particles 70 3.4 Creating Materials with a Desired Refraction Coefficient 71 3.5 Scattering by Small Particles Embedded in an Inhomogeneous Medium 72 3.6 Conclusions 72 References 73 4 Generalized Convex Functions and their Applications 77Adem Kiliçman andWedad Saleh 4.1 Brief Introduction 77 4.2 Generalized E-Convex Functions 78 4.3 E -Epigraph 84 4.4 Generalized s-Convex Functions 85 4.5 Applications to Special Means 96 References 98 5 Some Properties and Generalizations of the Catalan, Fuss, and Fuss-Catalan Numbers 101Feng Qi and Bai-Ni Guo 5.1 The Catalan Numbers 101 5.1.1 A Definition of the Catalan Numbers 101 5.1.2 The History of the Catalan Numbers 101 5.1.3 A Generating Function of the Catalan Numbers 102 5.1.4 Some Expressions of the Catalan Numbers 102 5.1.5 Integral Representations of the Catalan Numbers 103 5.1.6 Asymptotic Expansions of the Catalan Function 104 5.1.7 Complete Monotonicity of the Catalan Numbers 105 5.1.8 Inequalities of the Catalan Numbers and Function 106 5.1.9 The Bell Polynomials of the Second Kind and the Bessel Polynomials 109 5.2 The Catalan-Qi Function 111 5.2.1 The Fuss Numbers 111 5.2.2 A Definition of the Catalan-Qi Function 111 5.2.3 Some Identities of the Catalan-Qi Function 112 5.2.4 Integral Representations of the Catalan-Qi Function 114 5.2.5 Asymptotic Expansions of the Catalan-Qi Function 115 5.2.6 Complete Monotonicity of the Catalan-Qi Function 116 5.2.7 Schur-Convexity of the Catalan-Qi Function 118 5.2.8 Generating Functions of the Catalan-Qi Numbers 118 5.2.9 A Double Inequality of the Catalan-Qi Function 118 5.2.10 The q-Catalan-Qi Numbers and Properties 119 5.2.11 The Catalan Numbers and the k-Gamma and k-Beta Functions 119 5.2.12 Series Identities Involving the Catalan Numbers 119 5.3 The Fuss-Catalan Numbers 119 5.3.1 A Definition of the Fuss-Catalan Numbers 119 5.3.2 A Product-Ratio Expression of the Fuss-Catalan Numbers 120 5.3.3 Complete Monotonicity of the Fuss-Catalan Numbers 120 5.3.4 A Double Inequality for the Fuss-Catalan Numbers 121 5.4 The Fuss-Catalan-Qi Function 121 5.4.1 A Definition of the Fuss-Catalan-Qi Function 121 5.4.2 A Product-Ratio Expression of the Fuss-Catalan-Qi Function 122 5.4.3 Integral Representations of the Fuss-Catalan-Qi Function 123 5.4.4 Complete Monotonicity of the Fuss-Catalan-Qi Function 124 5.5 Some Properties for Ratios of Two Gamma Functions 124 5.5.1 An Integral Representation and Complete Monotonicity 125 5.5.2 An Exponential Expansion for the Ratio of Two Gamma Functions 125 5.5.3 A Double Inequality for the Ratio of Two Gamma Functions 125 5.6 Some NewResults on the Catalan Numbers 126 5.7 Open Problems 126 Acknowledgments 127 References 127 6 Trace Inequalities of Jensen Type for Self-adjoint Operators in Hilbert Spaces: A Survey of Recent Results 135Silvestru Sever Dragomir 6.1 Introduction 135 6.1.1 Jensen's Inequality 135 6.1.2 Traces for Operators in Hilbert Spaces 138 6.2 Jensen's Type Trace Inequalities 141 6.2.1 Some Trace Inequalities for Convex Functions 141 6.2.2 Some Functional Properties 145 6.2.3 Some Examples 151 6.2.4 More Inequalities for Convex Functions 154 6.3 Reverses of Jensen's Trace Inequality 157 6.3.1 A Reverse of Jensen's Inequality 157 6.3.2 Some Examples 163 6.3.3 Further Reverse Inequalities for Convex Functions 165 6.3.4 Some Examples 169 6.3.5 Reverses of Hölder's Inequality 174 6.4 Slater's Type Trace Inequalities 177 6.4.1 Slater's Type Inequalities 177 6.4.2 Further Reverses 180 References 188 7 Spectral Synthesis and Its Applications 193László Székelyhidi 7.1 Introduction 193 7.2 Basic Concepts and Function Classes 195 7.3 Discrete Spectral Synthesis 203 7.4 Nondiscrete Spectral Synthesis 217 7.5 Spherical Spectral Synthesis 219 7.6 Spectral Synthesis on Hypergroups 238 7.7 Applications 248 Acknowledgments 252 References 252 8 Various Ulam-Hyers Stabilities of Euler-Lagrange-Jensen General (a, b; k = <

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.

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