---
tag: BKZNC76RC8
kind: concept
name: Spherical function
msc: [43A90, "22E30"]
sources:
  - url: https://en.wikipedia.org/wiki/Zonal_spherical_function
same_as:
  wikidata: Q8073863
  wikipedia: Zonal spherical function
---

## Summary

A spherical function for a [[Gelfand pair|W38ZDXDVYC]] $(G, K)$ is a continuous function on $G$, invariant under $K$ on both sides.
It satisfies a product formula with an average over $K$.
Spherical functions are exactly the multiplicative functionals on the commutative algebra of $K$-bi-invariant functions.
They give the spherical transform, the analogue of the Fourier transform on the homogeneous space $G/K$.
Positive-definite spherical functions correspond to the irreducible unitary representations of $G$ with a nonzero $K$-fixed vector.

## Explanations

### L1 human

On the sphere, a zonal function depends only on the angle from the north pole.
Among such functions, the Legendre polynomials $P_\ell(\cos \theta)$ are special: averaging them over rotations behaves like multiplying exponentials.
These are the spherical functions of the pair of the rotation group and the rotations around one axis.
On the real line, with the trivial subgroup, the spherical functions are the exponentials $e^{s x}$.
They are the building blocks for analysis on spaces with a lot of symmetry.