Spherical harmonics

Spherical harmonics are orthogonal functions on the unit sphere that form angular solutions to Laplace’s equation. They provide a basis for representing functions and patterns across spherical surfaces.

Laplace’s equation: A partial differential equation stating that a function’s Laplacian is zero. Separation of variables in spherical coordinates produces spherical harmonics as its angular solutions.
Spherical coordinate system: A coordinate system that locates points using radius and two angles. Spherical harmonics describe angular variation at fixed radius in this coordinate system.
Geodesy: The science of measuring Earth’s shape, gravity field, and changing positions. Earth’s gravitational potential is commonly represented with spherical-harmonic coefficients.
Fourier transform: A transform that decomposes a function into frequency components using complex exponentials. It analyzes functions on flat or periodic domains rather than directly on a sphere.
Associated Legendre polynomials: Functions derived from Legendre polynomials that solve the polar part of separated angular equations. They supply the latitude-dependent factor in spherical harmonics.
Orthogonal functions: Functions whose pairwise inner products vanish under a specified integral or measure. Orthogonality lets spherical harmonics separate and quantify distinct spherical patterns.
Cosmic microwave background: Relic radiation from the early universe, observed across the sky as a nearly uniform microwave glow. Its temperature fluctuations are summarized by angular power across spherical-harmonic degrees.
Wavelet: A localized basis function used to analyze signals at multiple scales and positions. Wavelets retain spatial localization that global spherical harmonics generally lack.
Fourier series: An expansion of a periodic function into sums of sine and cosine waves. Spherical harmonics extend Fourier’s orthogonal-basis idea from a circle to a sphere.
Legendre polynomials: A sequence of polynomial solutions to Legendre’s differential equation. The zero-order spherical harmonics reduce to Legendre polynomials of the polar angle’s cosine.

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