The Laplace transform is an extremely versatile technique for solving differential equations, both ordinary and partial. It can also be used to solve difference equations. The present text, while mathematically rigorous, is readily accessible to students of either mathematics or engineering. Even the Dirac delta function, which is normally covered in a heuristic fashion, is given a completely justifiable treatment in the context of the Riemann-Stieltjes integral, yet at a level an undergraduate student can appreciate.

This is an easily accessible book on the approximation of functions---simple and without unnecessary details, but complete enough to include the main results of the theory. Except for a few sections, only functions of a real variable are treated. This work can be used as a textbook for graduate or advanced undergraduate courses or for self-study. Included in the volume are Notes at the end of each chapter, Problems, and a selected Bibliography.

Tauberian theory compares summability methods for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates. The author shows the development of the theory from the beginning and his expert commentary evokes the excitement surrounding the early results. He shows the fascination of the difficult Hardy-Littlewood theorems and of an unexpected simple proof, and extolls Wiener's breakthrough based on Fourier theory.

The theory of explicit formulas for regularized products and series forms a natural continuation of the analytic theory developed in LNM 1564. These explicit formulas can be used to describe the quantitative behavior of various objects in analytic number theory and spectral theory.

Though the search for good selectors dates back to the early twentieth century, selectors play an increasingly important role in current research. This book is the first to assemble the scattered literature into a coherent and elegant presentation of what is known and proven about selectors--and what remains to be found.

Presenting new results along with research spanning five decades, Fractional Cauchy Transforms provides a full treatment of the topic, from its roots in classical complex analysis to its current state. Self-contained, it includes introductory material and classical results, such as those associated with complex-valued measures on the unit circle, that form the basis of the developments that follow. The authors focus on concrete analytic questions, with functional analysis providing the general framework.

Infinitesimal analysis, once a synonym for calculus, is now viewed as a technique for studying the properties of an arbitrary mathematical object by discriminating between its standard and nonstandard constituents. Resurrected by A. Robinson in the early 1960's with the epithet 'nonstandard', infinitesimal analysis not only has revived the methods of infinitely small and infinitely large quantities, which go back to the very beginning of calculus, but also has suggested many powerful tools for research in every branch of modern mathematics.

Dedicated to the well-respected research mathematician Ambikeshwar Sharma, Frontiers in Interpolation and Approximation explores approximation theory, interpolation theory, and classical analysis.

This book is written primarily for undergraduates, though Part I may be judged by some teachers to be suitable for mathematicians during their last year at school. It includes the convergence theory that is commonly required for a university honours course in pure and applied mathematics, but excludes topics appropriate to post-graduate or to highly specialized courses of study. It has taken shape from sets of lectures I have given at various times during some fifteen years of university teaching.

This new approach to real analysis stresses the use of the subject in applications, by showing how the principles and theory of real analysis can be applied in a variety of settings in subjects ranging from Fourier series and polynomial approximation to discrete dynamical systems and nonlinear optimization. Users will be prepared for more intensive work in each topic through these applications and their accompanying exercises.

Carbohydrates offer a ready source of enantiomerically pure starting materials. They have been used for the imaginative synthesis of a wide range of compounds, and have been found to be effective chiral auxiliaries which enable the introduction of a range of functionalities in a highly enantioselective manner. In a subject dominated by volumes at research and professional level, this book provides a broad understanding of the use of carbohydrates in organic synthesis, at postgraduate student level.

This book is the first comprehensive textbook on the optical properties of photonic crystals. It deals not only with the properties of the radiation modes inside the crystals but also with their peculiar optical response to external fields. A general theory of linear and nonlinear optical response is given in a clear and detailed fashion using Green's function method. The symmetry of the eigenmodes is treated systematically using group theory to show how it affects the optical properties of photonic crystals.

Derivative spectrophotometry is a versatile tool for the quantitative estimation and analysis of substances. This book provides the theoretical background and the practical instructions necessary for using this highly efficient analytical technique. Drawing on carefully selected examples, the author points out the advantages and drawbacks of the various methods and reveals common pitfalls. Spectroscopists and analytical chemists in industry and academia will value this book as an indispensable guide to the use of derivative techniques in spectroscopy.

An outline of the modern theory of functional equations and inequalities in several variables. It consists of three parts. The first is devoted to additive and convex functions defined on linear spaces with semilinear topologies. In the second part, the problems of stability of functional equations in the sense of Ulam-Hyers-Rassias and in some function spaces are considered. In the last part, the functional equations in set-valued functions are dealt with, along with many results concerning those problems.

To a large extent, the development of non-linear analysis has been based on convexity: many existence theorems, in areas ranging from Fixed Point Theory to the Calculus of Variations, depend on this crucial assumption. Moreover, Convex Analysis, with its emphasis on epigraphs and subgradients, has been able to extend many classical results substituting for assumptions of smoothness (differentiability) assumptions of convex ty. In this sense convexity is also akin to non-smoothness.