Preface
This book is designed for researchers, graduate students, educators, and practitioners with an interest in mathematical analysis in particular and in mathematics in general. The book aims to present theory, methods, and applications of the chosen topics under several chapters that have recent research importance and use. Emphasis is made to present the basic developments concerning each idea in full detail, and also contain the most recent advances made in the corresponding area of study. The text is presented in a self-contained manner, providing at least an idea of the proof of all results, and giving enough references to enable the interested reader to follow subsequent studies in a still developing field. There are 20 selected chapters in this book and they are organized as follows.
The chapter "Spaces of Asymptotically Developable Functions and Applications" presents the functional structure of the spaces of asymptotically developable functions in several complex variables. The authors also illustrate the notion of summability with some applications concerning singularly perturbed systems of ordinary differential equations and Pfaffian systems.
The chapter "Duality for Gaussian Processes from Random Signed Measures" proves a number of results for a general class of Gaussian processes. Two features are stressed, first the Gaussian processes are indexed by a general measure space; second, the authors "adjust" the associated reproducing kernel Hilbert spaces (RKHSs) to the measurable category. Among other things, this allows us to give a precise necessary and sufficient condition for equivalence of a pair of probability measures (in sample space), which determine the corresponding two Gaussian processes.
In the chapter "Many-body Wave Scattering Problems for Small Scatterers and Creating Materials with a Desired Refraction Coefficient," formulas are derived for solutions of many-body wave scattering problems by small impedance particles embedded in an inhomogeneous medium. The limiting case is considered when the size a of small particles tends to zero while their number tends to infinity at a suitable rate. Equations for the limiting effective (self-consistent) field in the medium are derived. The theory is based on a study of integral equations and asymptotic of their solutions as a tends to zero. The case of wave scattering by many small particles embedded in an inhomogeneous medium is also studied. Applications of this theory to creating materials with a desired refraction coefficient are given. A recipe is given for creating such materials by embedding into a given material many small impedance particles with prescribed boundary impedances.
The chapter "Generalized Convex Functions and their Applications" focuses on convex functions and their generalization. The definition of convex function along with some relevant properties of such functions is given first, followed by a discussion on a simple geometric property. Then the e-convex function is generalized and some of their properties are established. Moreover, a generalized s-convex function is presented in the second sense and the paper presents some new inequalities of generalized Hermite-Hadamard's type for the class of functions whose second local fractional derivatives of order a in absolute value at certain powers are generalized s-convex functions in the second sense. At the end, some applications to special means are also presented.
The chapter "Some Properties and Generalizations of the Catalan, Fuss, and Fuss-Catalan Numbers" presents an expository review and survey for analytic generalizations and properties of the Catalan numbers, the Fuss numbers, the Fuss-Catalan numbers, the Catalan function, the Catalan-Qi function, the q-Catalan-Qi numbers, and the Fuss-Catalan-Qi function.
The chapter "Trace Inequalities of Jensen Type for Selfadjoint Operators in Hilbert Spaces: A Survey of Recent Results" provides a survey of recent results for trace inequalities related to the celebrated Jensen's and Slater's inequalities and their reverses. Applications for various functions of interest such as power and logarithmic functions are also emphasized. Trace inequalities for bounded linear operators in complex Hilbert spaces play an important role in Physics, in general, and in Quantum Mechanics, in particular.
The chapter "Spectral Synthesis and its Applications" presents a survey about spectral analysis and spectral synthesis. The chapter recalls the most important classical results in the field and in some cases new proofs for them are given. It also presents the most recent results in discrete, nondiscrete, and spherical spectral synthesis together with some applications.
The chapter "Various Ulam-Hyers Stabilities of Euler-Lagrange-Jensen General (a, b; k = a + b)-Sextic Functional Equations" elucidates the historical development of well-known stabilities of various types of functional equations such as quintic, sextic, septic, and octic functional equations. It introduced a new generalized Euler-Lagrange-Jensen sextic functional equation, obtained its general solution and further investigated its various fundamental stabilities and instabilities by having employed the famous Hyers' direct method as well as the alternative fixed point method. The chapter is expected to be helpful to analyze the stability of various functional equations applied in the physical sciences.
The chapter "A Note on the Split Common Fixed-Point Problem and its Variant Forms" proposed new algorithms for solving the split common fixed point problem and its variant forms, and prove the convergence results of the proposed algorithms. The split common fixed point problems have found its applications in various branches of mathematics both pure and applied. It provides a unified structure to study a large number of nonlinear mappings.
The chapter "Stabilities and Instabilities of Rational Functional Equations and Euler-Lagrange-Jensen (a, b)-Sextic Functional Equations" comprises the growth, importance and relevance of functional equations in other fields. Its fundamental and basic results of various stabilities are presented. The stability results of various rational and Euler-Lagrange-Jensen sextic functional equations are investigated. Application and geometrical interpretation of rational functional equation are also illustrated for the readers to study similar problems.
The chapter "Attractor of the Generalized Contractive Iterated Function System" deals with the problems to construct the fractal sets of the iterated function system for certain finite collection of F-contraction mappings defined on metric spaces as well as b-metric spaces. A new iterated function system called generalized F-contractive iterated function system is defined. Further, a method is presented to construct new fractals; where the resulting fractals are often self-similar but more general.
The chapter "Regular and Rapid Variations and some Applications" presents an overview of recent results on regular and rapid variations of functions and sequences and some their applications in selection principles theory, game theory, and asymptotic analysis of solutions of differential equations.
The chapter "n-Inner Products, n-Norms, and Angles Between Two Subspaces" discusses the concepts of n-inner products and n-norms for any natural number n, which are generalizations of the concepts of inner products and norms. It presents some geometric aspects of n-normed spaces and n-inner product spaces, especially regarding the notion of orthogonality and angles between two subspaces of such a space.
The chapter "Proximal Fiber Bundles on Nerve Complexes" introduces proximal fiber bundles of nerve complexes. Briefly, a nerve complex is a collection of filled triangles (2-simplexes) that have nonempty intersection. Nerve complexes are important in the study of shapes with a number of recent applications that include the classification of object shapes in digital images. The focus of this chapter is on fiber bundles defined by projections on a set of fibers that are nerve complexes into a base space such as the set of descriptions of nerve complexes. Two forms of fiber bundles are introduced, namely, spatial and descriptive, including a descriptive form BreMiller-Sloyer sheaf on a Vietoris-Rips complex. The introduction to nerve complexes includes a recent extension of nerve complexes that includes nerve spokes. A nerve spoke is a collection of filled triangles that always includes filled triangle in a nerve complex. A natural transition from the study of fibers that are nerve complexes is in the form of projections of pairs of fibers onto a local nervous system complex, which is a collection of nerve complexes that are glued together with spokes common to the nerve fibers. A number of results are given for fiber bundles viewed in the context of proximity spaces.
The chapter "Approximation by Generalizations of Hybrid Baskakov Type Operators Preserving Exponential Functions" deals with the approximation properties of the certain Baskakov-Szász operators. It estimates the results for these hybrid Baskakov-Szász type operators for exponential test functions. It also estimates a quantitative asymptotic formula for such operators. Mathematica software is used...
