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 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.

These proceedings were prepared in connection with the 14th International Conference on Approximation Theory, which was held April 7-10, 2013 in San Antonio, Texas. The conference was the fourteenth in a series of meetings in Approximation Theory held at various locations in the United States. The included invited and contributed papers cover diverse areas of approximation theory with a special emphasis on the most current and active areas such as compressed sensing, isogeometric analysis, anisotropic spaces, radial basis functions and splines. Classical and abstract approximation is also included. The book will be of interest to mathematicians, engineers\ and computer scientists working in approximation theory, computer-aided geometric design, numerical analysis and related application areas.

This book is the first to be devoted to the theory and applications of spherical (radial) basis functions (SBFs), which is rapidly emerging as one of the most promising techniques for solving problems where approximations are needed on the surface of a sphere. The aim of the book is to provide enough theoretical and practical details for the reader to be able to implement the SBF methods to solve real world problems. The authors stress the close connection between the theory of SBFs and that of the more well-known family of radial basis functions (RBFs), which are well-established tools for solving approximation theory problems on more general domains. The unique solvability of the SBF interpolation method for data fitting problems is established and an in-depth investigation of its accuracy is provided. Two chapters are devoted to partial differential equations (PDEs). One deals with the practical implementation of an SBF-based solution to an elliptic PDE and another which describes an SBF approach for solving a parabolic time-dependent PDE, complete with error analysis. The theory developed is illuminated with numerical experiments throughout. Spherical Radial Basis Functions, Theory and Applications will be of interest to graduate students and researchers in mathematics and related fields such as the geophysical sciences and statistics.

Advances in Maritime Technology and Engineering comprises a collection of the papers presented at the 7th International Conference on Maritime Technology and Engineering (MARTECH 2024) held in Lisbon, Portugal, on 14-16 May 2024. This Conference has evolved from the series of biannual national conferences in Portugal, which have become an international event, reflecting the internationalization of the maritime sector and its activities. MARTECH 2024 is the seventh of this new series of biannual conferences. This book comprises 142 contributions that were reviewed by an International Scientific Committee. Advances in Maritime Technology and Engineering is dedicated to maritime transportation, ports as well as maritime safety and reliability. It further comprises sections dedicated to ship design, cruise ship design, and to the structural aspects of ship design, such as ultimate strength and composites, subsea structures as pipelines, and to ship building and ship repair. The Proceedings in Marine Technology and Ocean Engineering series is dedicated to the publication of proceedings of peer-reviewed international conferences dealing with various aspects of “Marine Technology and Ocean Engineering”. The series includes the proceedings of the following conferences: the International Maritime Association of the Mediterranean (IMAM) conferences, the Marine Structures (MARSTRUCT) conferences, the Renewable Energies Offshore (RENEW) conferences and the Maritime Technology (MARTECH) conferences. The “Marine Technology and Ocean Engineering” series is also open to new conferences that cover topics on the sustainable exploration of marine resources in various fields, such as maritime transport and ports, usage of the ocean including coastal areas, nautical activities, the exploration and exploitation of mineral resources, the protection of the marine environment and is resources, and risk analysis, safety and reliability. The aim of the series is to stimulate advanced education and training through the wide dissemination of the results of scientific research.

A comprehensive introduction to basic operators of integral geometry and the relevant harmonic analysis for students and researchers.

The goal of this monograph is to develop the theory of wavelet harmonic analysis on the sphere. By starting with orthogonal polynomials and functional Hilbert spaces on the sphere, the foundations are laid for the study of spherical harmonics such as zonal functions. The book also discusses the construction of wavelet bases using special functions, especially Bessel, Hermite, Tchebychev, and Gegenbauer polynomials.