Article: Through the Kaleidoscope; Symmetries, Groups and Chebyshev Approximations from a Computational Point of View
Authors: H. Z. Munthe-Kaas, M. Nome, B. N. Ryland
Journal: To appear in Foundations of Computational Mathematics, proceedings of FoCM Budapest 2012.
Abstract: In this paper we survey parts of group theory, with emphasis on struc- tures that are important in design and analysis of numerical algorithms and in software design. In particular, we provide an extensive introduc- tion to Fourier analysis on locally compact abelian groups, and point to- wards applications of this theory in computational mathematics. Fourier analysis on non-commutative groups, with applications, is discussed more briefly. In the final part of the paper we provide an introduction to multi- variate Chebyshev polynomials. These are constructed by a kaleidoscope of mirrors acting upon an abelian group, and have recently been applied in numerical Clenshaw–Curtis type numerical quadrature and in spec- tral element solution of partial differential equations, based on triangular and simplicial subdivisions of the domain.
Hindsight notes: None yet.
5 Jan 2012