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  "body": "\n## Summary\n\nA spherical function for a [[Gelfand pair|W38ZDXDVYC]] $(G, K)$ is a continuous function on $G$, invariant under $K$ on both sides.\nIt satisfies a product formula with an average over $K$.\nSpherical functions are exactly the multiplicative functionals on the commutative algebra of $K$-bi-invariant functions.\nThey give the spherical transform, the analogue of the Fourier transform on the homogeneous space $G/K$.\nPositive-definite spherical functions correspond to the irreducible unitary representations of $G$ with a nonzero $K$-fixed vector.\n\n## Explanations\n\n### L1 human\n\nOn the sphere, a zonal function depends only on the angle from the north pole.\nAmong such functions, the Legendre polynomials $P_\\ell(\\cos \\theta)$ are special: averaging them over rotations behaves like multiplying exponentials.\nThese are the spherical functions of the pair of the rotation group and the rotations around one axis.\nOn the real line, with the trivial subgroup, the spherical functions are the exponentials $e^{s x}$.\nThey are the building blocks for analysis on spaces with a lot of symmetry.\n",
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