# Fast spin ±2 spherical harmonics transforms and application in cosmology

@article{Wiaux2007FastS, title={Fast spin ±2 spherical harmonics transforms and application in cosmology}, author={Yves Wiaux and Laurent Jacques and Pierre Vandergheynst}, journal={J. Comput. Phys.}, year={2007}, volume={226}, pages={2359-2371} }

## 31 Citations

FAST AND EXACT SPIN-s SPHERICAL HARMONIC TRANSFORMS

- Physics
- 2010

We demonstrate a fast spin-s spherical harmonic transform algorithm, which is flexible and exact for band-limited functions. In contrast to previous work, where spin transforms are computed…

A Mathematical View on Spin-Weighted Spherical Harmonics and Their Applications in Geodesy

- PhysicsMathematische Geodäsie/Mathematical Geodesy
- 2020

The spin-weighted spherical harmonics (by Newman and Penrose) form an orthonormal basis of L2(Ω) on the unit sphere Ω and have a huge field of applications. Mainly, they are used in quantum mechanics…

A pseudospectral matrix method for time-dependent tensor fields on a spherical shell

- PhysicsJ. Comput. Phys.
- 2013

Polarized wavelets and curvelets on the sphere

- Physics
- 2009

The statistics of the temperature anisotropies in the primordial cosmic microwave background radiation field provide a wealth of information for cosmology and for estimating cosmological parameters.…

A Novel Sampling Theorem on the Sphere

- Computer Science, MathematicsIEEE Transactions on Signal Processing
- 2011

This work develops a novel sampling theorem on the sphere and corresponding fast algorithms by associating the sphere with the torus through a periodic extension and highlights the advantages of the sampling theorem in the context of potential applications, notably in the field of compressive sampling.

A Unified Approach to Scalar, Vector, and Tensor Slepian Functions on the Sphere and Their Construction by a Commuting Operator

- MathematicsAnalysis and Applications
- 2022

Linear relationships between the spin-weighted and the classical scalar, vectorial, tensorial, and higher-rank spherical harmonics allow the construction of classical spherical-harmonic-based Slepian functions from their spin- Weighted counterparts, effectively rendering theConstruction of spherical-cap SlePian functions for any tensorial rank a computationally fast and numerically stable task.

Spin Needlets for Cosmic Microwave Background Polarization Data Analysis

- Physics
- 2008

Scalar wavelets have been used extensively in the analysis of cosmic microwave background (CMB) temperature maps. Spin needlets are a new form of (spin) wavelets which were introduced in the…

An Optimal-Dimensionality Sampling Scheme on the Sphere With Fast Spherical Harmonic Transforms

- MathematicsIEEE Transactions on Signal Processing
- 2014

We develop a sampling scheme on the sphere that permits accurate computation of the spherical harmonic transform and its inverse for signals band-limited at L using only L2 samples. We obtain the…

Improving the spatial dimensionality of Gauss-Legendre and equiangular sampling schemes on the sphere

- Mathematics2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
- 2017

An efficient GL sampling scheme with spatial dimensionality equal to that of equiangular scheme is proposed and it is demonstrated that the accuracy of the SHT is not affected with the proposed reduction in the spatialdimensionality.

## References

SHOWING 1-10 OF 69 REFERENCES

Computing Fourier Transforms and Convolutions on the 2-Sphere

- Computer Science, Mathematics
- 1994

Convolution theorems generalizing well known and useful results from the abelian case are used to develop a sampling theorem on the sphere, which reduces the calculation of Fourier transforms and convolutions of band-limited functions to discrete computations.

Correspondence Principle Between Spherical and Euclidean Wavelets, and Fast Directional Correlation on the Sphere With Steerable Filters

- Mathematics
- 2005

A fast algorithm is developed for the directional correlation of scalar band-limited signals and band-limited steerable filters on the sphere. The asymptotic complexity associated with it through…

Towards Safe and Effective High-Order Legendre Transforms with Applications to FFTs for the 2-sphere

- Computer ScienceAdv. Comput. Math.
- 2004

This study suggests a new approach to the high-order DLTs achieved by substitution of a simple alternative factorization of the associated Legendre functions in place of that used previously, which improves stability significantly for a wide range of useful problem sizes.

FFTs for the
2-Sphere-Improvements and
Variations

- Computer Science
- 1996

A reformulation and variation of the original algorithm is presented which results in a greatly improved inverse transform, and consequent improved convolution algorithm for such functions, which indicate that variations of the algorithm are both reliable and efficient for a large range of useful problem sizes.

An all sky analysis of polarization in the microwave background

- Physics
- 1997

Using the formalism of spin-weighted functions we present an all-sky analysis of polarization in the Cosmic Microwave Background (CMB). Linear polarization is a second-rank symmetric and traceless…

Spin‐s Spherical Harmonics and ð

- Mathematics
- 1967

Recent work on the Bondi‐Metzner‐Sachs group introduced a class of functions sYlm(θ, φ) defined on the sphere and a related differential operator ð. In this paper the sYlm are related to the…

CMB anisotropies: Total angular momentum method

- Physics
- 1997

of the Boltzmann equations brings out the geometric and model-independent aspects of temperature and polarization anisotropy formation. Large angle scalar polarization provides a robust means to…

Statistics of cosmic microwave background polarization

- Physics
- 1997

We present a formalism for analyzing a full-sky temperature and polarization map of the cosmic microwave background. Temperature maps are analyzed by expanding over the set of spherical harmonics to…

Cosmic microwave background polarization

- Physics
- 1995

Abstract Polarization of the cosmic microwave background, though not yet detected, provides a source of information about cosmological parameters complementary to temperature fluctuations. This paper…