Isotropic N-point basis functions and their properties
R. N. Cahn, Zachary Slepian
Abstract
Abstract Isotropic functions of positions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="bold">r</mml:mtext> </mml:mrow> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="bold">r</mml:mtext> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="bold">r</mml:mtext> </mml:mrow> </mml:mrow> <mml:mi>N</mml:mi> </mml:msub> </mml:math> , i.e. functions invariant under simultaneous rotations of all the coordinates, are conveniently formed using spherical harmonics and Clebsch–Gordan coefficients. An orthonormal basis of such functions provides a formalism suitable for analyzing isotropic distributions such as those that arise in cosmology, for instance in the clustering of galaxies as revealed by large-scale structure surveys. The algebraic properties of the basis functions are conveniently expressed in terms of 6- j and 9- j symbols. The calculation of relations among the basis functions is facilitated by ‘Yutsis’ diagrams for the addition and recoupling of angular momenta.