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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.

Engineering & Applied Sciences --- Civil & Environmental Engineering --- Operations Research --- Applied Mathematics --- Spherical harmonics. --- Spherical functions. --- Functions, Spherical --- Functions, Potential --- Potential functions --- Mathematics. --- Approximation theory. --- Integral equations. --- Partial differential equations. --- Special functions. --- Numerical analysis. --- Physics. --- Numerical Analysis. --- Special Functions. --- Approximations and Expansions. --- Integral Equations. --- Partial Differential Equations. --- Physics, general. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Mathematical analysis --- Special functions --- Partial differential equations --- Equations, Integral --- Functional equations --- Functional analysis --- Theory of approximation --- Functions --- Polynomials --- Chebyshev systems --- Math --- Science --- Spherical harmonics --- Transcendental functions --- Spheroidal functions --- Harmonic analysis --- Harmonic functions --- Functions, special. --- Differential equations, partial.

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The Augmented Spherical Wave (ASW) method is one of the most powerful approaches to handle the requirements of finite basis sets in DFT calculations. It is particularly suited for the calculation of the electronic, magnetic, and optical properties of solid-state materials. Recent developments allow application, in addition, to the elastic properties and phonon spectra. Due to the localized nature of the ASW basis set these properties can be easily interpreted in terms of atomic-like orbitals.   The book addresses all those who want to learn about methods for electronic structure calculations and the ASW method in particular.   This new edition has been thoroughly revised and extended. In particular, a chapter on the new, both very efficient and accurate spherical-wave based full potential ASW method has been added.

Spherical harmonics. --- Electronic structure. --- Density functionals. --- Differential equations, Partial. --- Density functional methods --- Density functional theory --- Functional methods, Density --- Functionals, Density --- Structure, Electronic --- Physics. --- Chemistry, Physical and theoretical. --- Condensed matter. --- Materials science. --- Condensed Matter Physics. --- Numerical and Computational Physics. --- Theoretical and Computational Chemistry. --- Materials Science, general. --- Partial differential equations --- Functional analysis --- Atomic structure --- Energy-band theory of solids --- Functions, Potential --- Potential functions --- Harmonic analysis --- Harmonic functions --- Chemistry. --- Materials. --- Numerical and Computational Physics, Simulation. --- Engineering --- Engineering materials --- Industrial materials --- Engineering design --- Manufacturing processes --- Physical sciences --- Materials --- Material science --- Chemistry, Theoretical --- Physical chemistry --- Theoretical chemistry --- Chemistry --- Natural philosophy --- Philosophy, Natural --- Dynamics --- Condensed materials --- Condensed media --- Condensed phase --- Materials, Condensed --- Media, Condensed --- Phase, Condensed --- Liquids --- Matter --- Solids