caa a22 4500 463248171 CHVBK 20180405153332.0 cr unu---uuuuu 170326e20070901xx s 000 0 eng 10.1007/s10455-006-9055-3 doi (NATIONALLICENCE)springer-10.1007/s10455-006-9055-3 Katz Gabriel Department of Mathematics, William Paterson University, 07470, Wayne, NJ, USA aut Deconstructing functions on quadratic surfaces into multipoles [Elektronische Daten] [Gabriel Katz] Any homogeneous polynomial P(x, y, z) of degree d, being restricted to a unit sphere S 2, admits essentially a unique representation of the form $$ \lambda + \sum_{k = 1}^d {\left[\prod_{j = 1}^k L_{kj}\right]} $$ where L kj 's are linear forms in x, y, and z and λ is a real number. The coefficients of these linear forms, viewed as 3D vectors, are called multipole vectors of P. In this paper, we consider similar multipole representations of polynomial and analytic functions on other quadratic surfaces Q(x, y, z) = c, real and complex. Over the complex numbers, the above representation is not unique, although the ambiguity is essentially finite. We investigate the combinatorics that depicts this ambiguity. We link these results with some classical theorems of harmonic analysis, theorems that describe decompositions of functions into sums of spherical harmonics. We extend these classical theorems (which rely on our understanding of the Laplace operator $$\Delta_{S^2}$$ ) to more general differential operators Δ Q that are constructed with the help of the quadratic form Q(x, y, z). Then we introduce modular spaces of multipoles. We study their intricate geometry and topology using methods of algebraic geometry and singularity theory. The multipole spaces are ramified over vector or projective spaces, and the compliments to the ramification sets give rise to a rich family of K(π, 1)-spaces, where π runs over a variety of modified braid groups. Springer Science + Business Media B.V., 2007 Annals of Global Analysis and Geometry Springer Netherlands 32/2(2007-09-01), 167-207 0232-704X 32:2<167 2007 32 10455 https://doi.org/10.1007/s10455-006-9055-3 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10455-006-9055-3 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Katz Gabriel Department of Mathematics, William Paterson University, 07470, Wayne, NJ, USA aut NATIONALLICENCE 773 0- Annals of Global Analysis and Geometry Springer Netherlands 32/2(2007-09-01), 167-207 0232-704X 32:2<167 2007 32 10455 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer