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Month: May 2021

The Funk-Hecke formula

Leopold Gegenbauer (1849 - 1903)
Leopold Gegenbauer (1849 - 1903) famous in particular for his orthogonal polynomials.

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,yRn we write xy=x1y1++xnyn and |x|=xx=x21++x2n.

Sphere. Let σSn1 be the uniform probability measure on the unit sphere of Rn, n2,

Sn1={xRn:|x|=1}.

Denoting dx the trace of the Lebesgue measure on Sn1, we have

σSn1(dx)=dx|Sn1|where|Sn1|=Sn1dx=2πn2Γ(n2)

is the surface area of Sn1. If n=2 we recover the perimeter 2π of the unit circle.

If n=1, then Sn1=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 12δ1+12δ1, which is 12 times the counting measure δ1+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 ySn1,

f(xy)σSn1(dx)=Γ(n2)πΓ(n12)11f(t)(1t2)n32dt.

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

11(1t2)n32dt=10(1u)n32u12du=Beta(n12,12)=Γ(12)Γ(n12)Γ(n2).

To prove the Funk-Hecke formula, we can slice the integration over Sn1 with respect to the value of t=cos(θ)=xy[1,1], θ[0,π]. We note that

dt=sin(θ)dθ=1t2dθ.

Moreover the intersection of Sn1 with the hyperplane {xRn:xy=t} is a sphere Sn2t of dimension n2 and of radius 1t2 (a couple of points if n=2). Its surface area is |Sn2t|=1t2n2|Sn2| (equal to 2 if n=2). As a consequence, we get

Sn1f(xy)dx=π0f(t)|Sn2t|dθ=|Sn2|11f(t)(1t2)n32dt.

It remains to divide both sides by |Sn1| and to observe that we have |Sn2||Sn1|=Γ(n2)πΓ(n12).

Probability laws. In probabilistic terms, if X is a random vector of Rn uniformly distributed on Sn1 then for all ySn1, the law of Xy has density

t[1,1]Γ(n2)πΓ(n12)(1t2)n32.

This is the Beta law on [1,1] with parameters a=b=n+14.
This law does not depend on the choice of ySn1.
It is symmetric in the sense that Xy and (Xy) have same law.
The law of |Xy| is the image of the law Beta(12,n12) by the map uu.
The law of Xy is in particular:

  • if n=2, an arcsine law with density 1t[1,1]π1t2,
  • if n=3, a uniform law with density 1t[1,1]2 (Archimedes principle, see below),
  • if n=4, a semicircle law with density 21t21t[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 Sn1 then the projection (X1,,Xn2) is uniformly distributed on the unit ball of Rn2. What matters is to lose two dimensions.

It does not work if we replace n2 by n1. 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 {xR2:|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

tΓ(α+1)πΓ(α+12)(1t2)α12,

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=n22. 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 Sn1. We identify each of them to its restriction to Sn1. These polynomials are orthogonal with respect to the uniform probability measure on Sn1, 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=n22, normalized in such a way that Pk(1)=1, can be expressed in terms of spherical harmonics of degree k. More precisely, if (Yk,)1Zn,k are all the spherical harmonics on Sn1 of degree k and if Zn,k is their number, then the addition formula states that for all x,ySn1,

Zn,k=1Yk,(x)Yk,(y)=Zn,kPk(xy).

General Funk-Hecke formula. It reads, for any bounded measurable f:Sn1R, any ySn1, and any spherical harmonics Yk on Sn1 of degree k0,

f(xy)Yk(x)σSn1(dx)=λkYk(y)

where

λk=Γ(n2)πΓ(n12)11f(t)Pk(t)(1t2)n32dt.

We recover the simple Funk-Hecke formula by taking k=0 for which Y0=1 and P0=1.

In other words, for any kernel'' xf(xy) on Sn1, a spherical harmonics Yk of degree k is an eigenfunction, associated to the eigenvalue λk, of the Funk transform

gf(x)g(x)σSn1(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.

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