
The Funk-Hecke formula has an analytic, geometric, and probabilistic content. In its simplest form, and probabilistically, it gives the law of the projection of the uniform law on the sphere on any diameter of the sphere. It allows dimension reduction of multivariate integrals.
For x,y∈Rn we write x⋅y=x1y1+⋯+xnyn and |x|=√x⋅x=√x21+⋯+x2n.
Sphere. Let σSn−1 be the uniform probability measure on the unit sphere of Rn, n≥2,
Sn−1={x∈Rn:|x|=1}.
Denoting dx the trace of the Lebesgue measure on Sn−1, we have
σSn−1(dx)=dx|Sn−1|where|Sn−1|=∫Sn−1dx=2πn2Γ(n2)
is the surface area of Sn−1. If n=2 we recover the perimeter 2π of the unit circle.
If n=1, then Sn−1=S0={−1,1} is a couple of points, and we can define the uniform probability measure on it as being the Rademacher or symmetric Bernoulli distribution 12δ−1+12δ1, which is 12 times the counting measure δ−1+δ1 which plays the role of the trace of the Lebesgue measure, and from this point of view the surface area is 2, which is exactly the value given by the Gamma based formula above when n=1.
Funk-Hecke formula. In its basic form, the Funk-Hecke formula states that for all bounded measurable f:[−1,1]↦R and all y∈Sn−1,
∫f(x⋅y)σSn−1(dx)=Γ(n2)√πΓ(n−12)∫1−1f(t)(1−t2)n−32dt.
The formula does not depend on y, an invariance due to spherical symmetry.
The constant in the right hand side can be easily recovered by taking f=1, indeed
∫1−1(1−t2)n−32dt=∫10(1−u)n−32u−12du=Beta(n−12,12)=Γ(12)Γ(n−12)Γ(n2).
To prove the Funk-Hecke formula, we can slice the integration over Sn−1 with respect to the value of t=cos(θ)=x⋅y∈[−1,1], θ∈[0,π]. We note that
dt=sin(θ)dθ=√1−t2dθ.
Moreover the intersection of Sn−1 with the hyperplane {x∈Rn:x⋅y=t} is a sphere Sn−2t of dimension n−2 and of radius √1−t2 (a couple of points if n=2). Its surface area is |Sn−2t|=√1−t2n−2|Sn−2| (equal to 2 if n=2). As a consequence, we get
∫Sn−1f(x⋅y)dx=∫π0f(t)|Sn−2t|dθ=|Sn−2|∫1−1f(t)(1−t2)n−32dt.
It remains to divide both sides by |Sn−1| and to observe that we have |Sn−2||Sn−1|=Γ(n2)√πΓ(n−12).
Probability laws. In probabilistic terms, if X is a random vector of Rn uniformly distributed on Sn−1 then for all y∈Sn−1, the law of X⋅y has density
t∈[−1,1]↦Γ(n2)√πΓ(n−12)(1−t2)n−32.
This is the Beta law on [−1,1] with parameters a=b=n+14.
This law does not depend on the choice of y∈Sn−1.
It is symmetric in the sense that X⋅y and −(X⋅y) have same law.
The law of |X⋅y| is the image of the law Beta(12,n−12) by the map u↦√u.
The law of X⋅y is in particular:
- if n=2, an arcsine law with density 1t∈[−1,1]π√1−t2,
- if n=3, a uniform law with density 1t∈[−1,1]2 (Archimedes principle, see below),
- if n=4, a semicircle law with density 2√1−t21t∈[−1,1]π.
Archimedes principle. It states that the projection of the uniform law of the unit sphere of R3 on a diameter is the uniform law on the diameter. It is the case n=3 of the Funk-Hecke formula above. More generally, if (X1,…,Xn) is a random vector of Rn uniformly distributed on the unit sphere Sn−1 then the projection (X1,…,Xn−2) is uniformly distributed on the unit ball of Rn−2. What matters is to lose two dimensions.
It does not work if we replace n−2 by n−1. To see it, let us recall that the arcsine law on [−1,1] is the projection of the uniform law on S1, which is the unit sphere of R2, while the semicircle law on [−1,1] is the projection of the uniform law on the disc {x∈R2:|x|≤1} which is the unit ball of R2. Also it follows from the Archimedes principle that the projection of the uniform law on S2 on a plane containing the origin cannot be the uniform law on a disc. Indeed, if it was the case, the projection on a diameter would be the semicircle law, while the Archimedes principle states that it is the uniform law.
Gegenbauer or ultraspherical orthogonal polynomials. This family is orthogonal with respect to the probability measure on [−1,1] with density
t↦Γ(α+1)√πΓ(α+12)(1−t2)α−12,
where α is a parameter. We could call this distribution the Gegenbauer law. The law that appear in the Funk-Hecke formula corresponds to the case α=n−22. It is the Beta law on [−1,1] with parameters a=b=2α+34. The Gegenbauer polynomials form a special class of Jacobi polynomials, and include Chebyshev T, Legendre, and Chebyshev U polynomials as the special cases n=2, n=3, and n=4.
Spherical harmonics. They are by definition the homogeneous polynomials of n variables which are harmonic on Rn. Since they are homogeneous, they are characterized by their restriction to Sn−1. We identify each of them to its restriction to Sn−1. These polynomials are orthogonal with respect to the uniform probability measure on Sn−1, and we normalize them in such a way that they form an orthonormal sequence. It turns out moreover that the spherical harmonics form the eigenfunctions of the Laplace-Beltrami operator on the sphere.
The k-th Gegenbauer polynomial Pk of parameter α=n−22, normalized in such a way that Pk(1)=1, can be expressed in terms of spherical harmonics of degree k. More precisely, if (Yk,ℓ)1≤ℓ≤Zn,k are all the spherical harmonics on Sn−1 of degree k and if Zn,k is their number, then the addition formula states that for all x,y∈Sn−1,
Zn,k∑ℓ=1Yk,ℓ(x)Yk,ℓ(y)=Zn,kPk(x⋅y).
General Funk-Hecke formula. It reads, for any bounded measurable f:Sn−1→R, any y∈Sn−1, and any spherical harmonics Yk on Sn−1 of degree k≥0,
∫f(x⋅y)Yk(x)σSn−1(dx)=λkYk(y)
where
λk=Γ(n2)√πΓ(n−12)∫1−1f(t)Pk(t)(1−t2)n−32dt.
We recover the simple Funk-Hecke formula by taking k=0 for which Y0=1 and P0=1.
In other words, for any kernel'' x↦f(x⋅y) on Sn−1, a spherical harmonics Yk of degree k is an eigenfunction, associated to the eigenvalue λk, of the Funk transform
g↦∫f(x⋅∙)g(x)σSn−1(dx).
Naming. The Funk-Hecke formula is named after the mathematicians Paul Funk (1886 - 1969) and Erich Hecke (1887 - 1947), two former students of David Hilbert, for their work in harmonic analysis published in 1916 and 1918 respectively.
Stochastic processes. The uniform law on the sphere is invariant for spherical Brownian motion. How about the law of the projection of spherical Brownian motion on a diameter? It turns out that this is again a Markov diffusion process for which the Gegenbauer law is invariant, related to Wright-Fisher diffusions and Jacobi operators.
Further reading.
- Discrete energy on rectifiable sets (Chapter 5)
Borodachov, Hardin, and Saff - Springer Monographs in Mathematics (2019) Beta laws\ On this blog - 2011 - Central limit theorem for convex bodies
On this blog - 2011 - Spherical Harmonics
Müller - Springer Lecture Notes in Mathematics 17 (1996) - Remarques sur les semigroupes de Jacobi
Bakry - Astérisque 236 23-39 (1996) - Archimedes, Gauss, and Stein
Pitman and Ross - Notices Amer. Math. Soc. 59(10) 1416-1421 (2012) - Projections of spherical Brownian motion
Mijatović, Mramor, and Bravo - Electron. Commun. Probab. 23 1-12 (2018) - Beiträge zur Theorie der Kugelfunktionen
Funk - Math. Ann. 77 136-152 (1916) - Uber orthogonal-invariante Integralgleichungen
Hecke - Math. Ann. 78 398-404 (1918)