Mathematical Analysis and Applications

Selected Topics

Michael Ruzhansky (Redaktør) ; Hemen Dutta (Redaktør) ; Ravi P. Agarwal (Redaktør)

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. Les mer
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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.

Fakta

Innholdsfortegnelse

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 Fernandez


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-Body Wave Scattering Problems for Small Scatterers and Creating Materials with a Desired Refraction Coefficient 57
Alexander G. Ramm


3.1 Introduction 57


3.2 Derivation of the Formulas for One-Body Wave 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 Kilicman and Wedad 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 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


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