Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere

Dennis, Mark R.

doi:10.1088/0305-4470/37/40/011

Cited by 42 publications

(68 citation statements)

References 21 publications

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“…The ℓ-th multipole of the CMB, T ℓ , can, instead of being expanded in spherical harmonics, be written uniquely [18,40,41] in terms of a scalar A (ℓ) which depends only on the total power in this multipole and ℓ unit vectors {v (ℓ,i) |i = 1, ..., ℓ}. These "multipole vectors" encode all the information about the phase relationships of the a ℓm .…”

Section: Multipole Vectorsmentioning

confidence: 99%

Uncorrelated universe: Statistical anisotropy and the vanishing angular correlation function in WMAP years 1–3

et al. 2007

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The large-angle (low-ℓ) correlations of the Cosmic Microwave Background (CMB) as reported by the Wilkinson Microwave Anisotropy Probe (WMAP) after their first year of observations exhibited statistically significant anomalies compared to the predictions of the standard inflationary big-bang model. We suggested then that these implied the presence of a solar system foreground, a systematic correlated with solar system geometry, or both. We re-examine these anomalies for the data from the first three years of WMAP's operation. We show that, despite the identification by the WMAP team of a systematic correlated with the equinoxes and the ecliptic, the anomalies in the firstyear Internal Linear Combination (ILC) map persist in the three-year ILC map, in all-but-one case at similar statistical significance. The three-year ILC quadrupole and octopole therefore remain inconsistent with statistical isotropy -they are correlated with each other (99.6%C.L.), and there are statistically significant correlations with local geometry, especially that of the solar system. The angular two-point correlation function at scales > 60 degrees in the regions outside the (kp0) galactic cut, where it is most reliably determined, is approximately zero in all wavebands and is even more discrepant with the best fit ΛCDM inflationary model than in the first-year data -99.97%C.L. for the new ILC map. The full-sky ILC map, on the other hand, has a non-vanishing angular two-point correlation function, apparently driven by the region inside the cut, but which does not agree better with ΛCDM. The role of the newly identified low-ℓ systematics is more puzzling than reassuring.

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Section: Multipole Vectorsmentioning

confidence: 99%

Uncorrelated universe: Statistical anisotropy and the vanishing angular correlation function in WMAP years 1–3

et al. 2007

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“…Classical harmonic analysis on spheres provides an alternative approach to function deconstructions on quadratic surfaces [see formulas (6) and (8)]. We observe that these deconstructions, based on spherical harmonics, behave in a natural way under complex linear change of coordinates that preserve a given non-degenerated quadratic form Q(v) = vB, v .…”

Section: In the Space Of Real Continuous Functions F On S 2 There Is mentioning

confidence: 96%

“…The direct summands in (6) are orthogonal with respect to the inner product in V(d; R) defined by the formula f , g = S 2 f · g dm. The measure dm on the sphere S 2 is the standard one.…”

Section: Corollarymentioning

confidence: 99%

Deconstructing functions on quadratic surfaces into multipoles

Katz

2007

Ann Glob Anal Geom

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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, where L kj 's are linear forms in x, y and z and λ k 's are real numbers. 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 ∆ S 2 ) to more general differential operators ∆ 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.

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“…A general spin s state |ψ is represented in the present complex characterization by the so called Majorana function [13,14]:…”

Section: Complex Holomorphic Characterization: Majorana Functionsmentioning

confidence: 99%

Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

Calixto

Guerrero

Sánchez-Monreal

2008

J Fourier Anal Appl

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Using coherent-state techniques, we prove a sampling theorem for Majorana's (holomorphic) functions on the Riemann sphere and we provide an exact reconstruction formula as a convolution product of N samples and a given reconstruction kernel (a sinc-type function). We also discuss the effect of over-and undersampling. Sample points are roots of unity, a fact which allows explicit inversion formulas for resolution and overlapping kernel operators through the theory of Circulant Matrices and Rectangular Fourier Matrices. The case of band-limited functions on the Riemann sphere, with spins up to J , is also considered. The connection with the standard Euler angle picture, in terms of spherical harmonics, is established through a discrete Bargmann transform. Mathematics Subject Classification (2000)32A10 · 42B05 · 94A12 · 94A20 · 81R30 Communicated by Thomas Strohmer.M. Calixto ( ) · J.C. Sánchez-Monreal

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Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere

Cited by 42 publications

References 21 publications

Uncorrelated universe: Statistical anisotropy and the vanishing angular correlation function in WMAP years 1–3

Uncorrelated universe: Statistical anisotropy and the vanishing angular correlation function in WMAP years 1–3

Deconstructing functions on quadratic surfaces into multipoles

Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

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