Preface xv ; About the Editors xxi ; List of Contributors xxiii ; 1 Spaces of Asymptotically Developable Functions and Applications 1 / Sergio 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 23 / Palle 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 57 / Alexander 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 77 / Adem 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 101 / Feng 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 135 / Silvestru 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 193 / László 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 ;