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The first book to provide a systematic account of recent developments and applications in harmonic approximation, progresses from classical results concerning uniform approximation on compact sets through fusion techniques to deal with approximation on unbounded sets.
This monograph records progress in approximation theory and harmonic analysis on balls and spheres, and presents contemporary material that will be useful to analysts in this area. While the first part of the book contains mainstream material on the subject, the second and the third parts deal with more specialized topics, such as analysis in weight spaces with reflection invariant weight functions, and analysis on balls and simplexes. The last part of the book features several applications, including cubature formulas, distribution of points on the sphere, and the reconstruction algorithm in computerized tomography. This book is directed at researchers and advanced graduate students in analysis. Mathematicians who are familiar with Fourier analysis and harmonic analysis will understand many of the concepts that appear in this manuscript: spherical harmonics, the Hardy-Littlewood maximal function, the Marcinkiewicz multiplier theorem, the Riesz transform, and doubling weights are all familiar tools to researchers in this area.
This volume presents the proceedings of a conference on Harmonic Analysis and Number Theory held at McGill University (Montreal) in April 1996. The papers are dedicated to the memory of Carl Herz, who had deep interests in both harmonic analysis and number theory. These two disciplines have a symbiotic relationship that is reflected in the papers in this book.
These notes provide an introduction to the theory of spherical harmonics in an arbitrary dimension as well as an overview of classical and recent results on some aspects of the approximation of functions by spherical polynomials and numerical integration over the unit sphere. The notes are intended for graduate students in the mathematical sciences and researchers who are interested in solving problems involving partial differential and integral equations on the unit sphere, especially on the unit sphere in three-dimensional Euclidean space. Some related work for approximation on the unit disk in the plane is also briefly discussed, with results being generalizable to the unit ball in more dimensions.
This concisely written book gives an elementary introduction to a classical area of mathematics – approximation theory – in a way that naturally leads to the modern field of wavelets. The exposition, driven by ideas rather than technical details and proofs, demonstrates the dynamic nature of mathematics and the influence of classical disciplines on many areas of modern mathematics and applications. Featuring classical, illustrative examples and constructions, exercises, and a discussion of the role of wavelets to areas such as digital signal processing and data compression, the book is one of the few to describe wavelets in words rather than mathematical symbols.
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This graduate textbook describes atomic-level kinetics (mechanisms and rates) of thermal energy storage, transport (conduction, convection, and radiation), and transformation (various energy conversions) by principal energy carriers. The approach combines the fundamentals of molecular orbitals-potentials, statistical thermodynamics, computational molecular dynamics, quantum energy states, transport theories, solid-state and fluid-state physics, and quantum optics. The textbook presents a unified theory, over fine-structure/molecular-dynamics/Boltzmann/macroscopic length and time scales, of heat transfer kinetics in terms of transition rates and relaxation times, and its modern applications, including nano- and microscale size effects. Numerous examples, illustrations, and homework problems with answers that enhance learning are included. This new edition includes applications in energy conversion (including chemical bond, nuclear, and solar), expanded examples of size effects, inclusion of junction quantum transport, and discussion of graphene and its phonon and electronic conductances. New appendix coverage of Phonon Contributions Seebeck Coefficient and Monte Carlo Methods are also included.
'The third edition of this book is designed to carefully and coherently introduce fractional calculus to physicists, by applying the ideas to two distinct applications: classical problems and multi-particle quantum problems. There remain many open questions and the field remains an active area of research. Dr Herrmann’s book is an excellent introduction to this field of study.'Contemporary PhysicsThe book presents a concise introduction to the basic methods and strategies in fractional calculus which enables the reader to catch up with the state-of-the-art in this field and to participate and contribute in the development of this exciting research area.This book is devoted to the application of fractional calculus on physical problems. The fractional concept is applied to subjects in classical mechanics, image processing, folded potentials in cluster physics, infrared spectroscopy, group theory, quantum mechanics, nuclear physics, hadron spectroscopy up to quantum field theory and will surprise the reader with new intriguing insights.This new, extended edition includes additional chapters about numerical solution of the fractional Schrödinger equation, self-similarity and the geometric interpretation of non-isotropic fractional differential operators. Motivated by the positive response, new exercises with elaborated solutions are added, which significantly support a deeper understanding of the general aspects of the theory.Besides students as well as researchers in this field, this book will also be useful as a supporting medium for teachers teaching courses devoted to this subject.
Recent years have seen a growing interest in the field of thermodynamic properties of solids due to the development of advanced experimental and modeling tools. Predicting structural phase transitions and thermodynamic properties find important applications in condensed matter and materials science research, as well as in interdisciplinary research involving geophysics and Earth Sciences. The present edited book, with contributions from leading researchers around the world, is aimed to meet the need of academic and industrial researchers, graduate students and non-specialists working in these fields. The book covers various experimental and theoretical techniques relevant to the subject.