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 \( {x,y\in\mathbb{R}^n} \) we write \( {x\cdot y=x_1y_1+\cdots+x_ny_n} \) and \( {|x|=\sqrt{x\cdot x}=\sqrt{x_1^2+\cdots+x_n^2}} \).
Sphere. Let \( {\sigma_{\mathbb{S}^{n-1}}} \) be the uniform probability measure on the unit sphere of \( {\mathbb{R}^n} \), \( {n\geq2} \),
\[ \mathbb{S}^{n-1}=\{x\in\mathbb{R}^n:|x|=1\}. \]
Denoting \( {\mathrm{d}x} \) the trace of the Lebesgue measure on \( {\mathbb{S}^{n-1}} \), we have
\[ \sigma_{\mathbb{S}^{n-1}}(\mathrm{d}x) =\frac{\mathrm{d}x}{|\mathbb{S}^{n-1}|} \quad\text{where}\quad |\mathbb{S}^{n-1}|=\int_{\mathbb{S}^{n-1}}\mathrm{d}x=2\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})} \]
is the surface area of \( {\mathbb{S}^{n-1}} \). If \( {n=2} \) we recover the perimeter \( {2\pi} \) of the unit circle.
If \( {n=1} \), then \( {\mathbb{S}^{n-1}=\mathbb{S}^0=\{-1,1\}} \) is a couple of points, and we can define the uniform probability measure on it as being the Rademacher or symmetric Bernoulli distribution \( {\frac{1}{2}\delta_{-1}+\frac{1}{2}\delta_1} \), which is \( {\frac{1}{2}} \) times the counting measure \( {\delta_{-1}+\delta_1} \) 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 \( {f:[-1,1]\mapsto\mathbb{R}} \) and all \( {y\in\mathbb{S}^{n-1}} \),
\[ \int f(x\cdot y)\sigma_{\mathbb{S}^{n-1}}(\mathrm{d}x)=\frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}\Gamma(\frac{n-1}{2})}\int_{-1}^1f(t)(1-t^2)^{\frac{n-3}{2}}\mathrm{d}t. \]
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
\[ \int_{-1}^1(1-t^2)^{\frac{n-3}{2}}\mathrm{d}t =\int_0^1(1-u)^{\frac{n-3}{2}}u^{-\frac{1}{2}}\mathrm{d}u =\mathrm{Beta}\Bigr(\frac{n-1}{2},\frac{1}{2}\Bigr) =\frac{\Gamma(\frac{1}{2})\Gamma(\frac{n-1}{2})}{\Gamma(\frac{n}{2})}. \]
To prove the Funk-Hecke formula, we can slice the integration over \( {\mathbb{S}^{n-1}} \) with respect to the value of \( {t=\cos(\theta)=x\cdot y\in[-1,1]} \), \( {\theta\in[0,\pi]} \). We note that
\[ \mathrm{d}t=\sin(\theta)\mathrm{d}\theta=\sqrt{1-t^2}\mathrm{d}\theta. \]
Moreover the intersection of \( {\mathbb{S}^{n-1}} \) with the hyperplane \( {\{x\in\mathbb{R}^n:x\cdot y=t\}} \) is a sphere \( {S^{n-2}_t} \) of dimension \( {n-2} \) and of radius \( {\sqrt{1-t^2}} \) (a couple of points if \( {n=2} \)). Its surface area is \( {|S^{n-2}_t|=\sqrt{1-t^2}^{n-2}|\mathbb{S}^{n-2}|} \) (equal to \( {2} \) if \( {n=2} \)). As a consequence, we get
\[ \int_{\mathbb{S}^{n-1}} f(x\cdot y)\mathrm{d}x =\int_0^\pi f(t)|S^{n-2}_t|\mathrm{d}\theta =|\mathbb{S}^{n-2}|\int_{-1}^1f(t)(1-t^2)^{\frac{n-3}{2}}\mathrm{d}t. \]
It remains to divide both sides by \( {|\mathbb{S}^{n-1}|} \) and to observe that we have \( {\frac{|\mathbb{S}^{n-2}|}{|\mathbb{S}^{n-1}|}=\frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}\Gamma(\frac{n-1}{2})}} \).
Probability laws. In probabilistic terms, if \( {X} \) is a random vector of \( {\mathbb{R}^n} \) uniformly distributed on \( {\mathbb{S}^{n-1}} \) then for all \( {y\in\mathbb{S}^{n-1}} \), the law of \( {X\cdot y} \) has density
\[ t\in[-1,1]\mapsto \frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}\Gamma(\frac{n-1}{2})} (1-t^2)^{\frac{n-3}{2}}. \]
This is the Beta law on \( {[-1,1]} \) with parameters \( {a=b=\frac{n+1}{4}} \).
This law does not depend on the choice of \( {y\in\mathbb{S}^{n-1}} \).
It is symmetric in the sense that \( {X\cdot y} \) and \( {-(X\cdot y)} \) have same law.
The law of \( {|X\cdot y|} \) is the image of the law \( {\mathrm{Beta}\left(\frac{1}{2},\frac{n-1}{2}\right)} \) by the map \( {u\mapsto\sqrt{u}} \).
The law of \( {X\cdot y} \) is in particular:
- if \( {n=2} \), an arcsine law with density \( {\frac{\mathbf{1}_{t\in[-1,1]}}{\pi\sqrt{1-t^2}}} \),
- if \( {n=3} \), a uniform law with density \( {\frac{\mathbf{1}_{t\in[-1,1]}}{2}} \) (Archimedes principle, see below),
- if \( {n=4} \), a semicircle law with density \( {\frac{2\sqrt{1-t^2}\mathbf{1}_{t\in[-1,1]}}{\pi}} \).
Archimedes principle. It states that the projection of the uniform law of the unit sphere of \( {\mathbb{R}^3} \) 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 \( {(X_1,\ldots,X_n)} \) is a random vector of \( {\mathbb{R}^n} \) uniformly distributed on the unit sphere \( {\mathbb{S}^{n-1}} \) then the projection \( {(X_1,\ldots,X_{n-2})} \) is uniformly distributed on the unit ball of \( {\mathbb{R}^{n-2}} \). 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 \( {\mathbb{S}^1} \), which is the unit sphere of \( {\mathbb{R}^2} \), while the semicircle law on \( {[-1,1]} \) is the projection of the uniform law on the disc \( {\{x\in\mathbb{R}^2:|x|\leq 1\}} \) which is the unit ball of \( {\mathbb{R}^2} \). Also it follows from the Archimedes principle that the projection of the uniform law on \( {\mathbb{S}^2} \) 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
\[ t\mapsto\frac{\Gamma(\alpha+1)}{\sqrt{\pi}\Gamma(\alpha+\frac{1}{2})}(1-t^2)^{\alpha-\frac{1}{2}}, \]
where \( {\alpha} \) is a parameter. We could call this distribution the Gegenbauer law. The law that appear in the Funk-Hecke formula corresponds to the case \( {\alpha=\frac{n-2}{2}} \). It is the Beta law on \( {[-1,1]} \) with parameters \( {a=b=\frac{2\alpha+3}{4}} \). 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 \( {\mathbb{R}^n} \). Since they are homogeneous, they are characterized by their restriction to \( {\mathbb{S}^{n-1}} \). We identify each of them to its restriction to \( {\mathbb{S}^{n-1}} \). These polynomials are orthogonal with respect to the uniform probability measure on \( {\mathbb{S}^{n-1}} \), 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 \( {P_k} \) of parameter \( {\alpha=\frac{n-2}{2}} \), normalized in such a way that \( {P_k(1)=1} \), can be expressed in terms of spherical harmonics of degree \( {k} \). More precisely, if \( {{(Y_{k,\ell})}_{1\leq\ell\leq Z_{n,k}}} \) are all the spherical harmonics on \( {\mathbb{S}^{n-1}} \) of degree \( {k} \) and if \( {Z_{n,k}} \) is their number, then the addition formula states that for all \( {x,y\in\mathbb{S}^{n-1}} \),
\[ \sum_{\ell=1}^{Z_{n,k}}Y_{k,\ell}(x)Y_{k,\ell}(y)=Z_{n,k}P_k(x\cdot y). \]
General Funk-Hecke formula. It reads, for any bounded measurable \( {f:\mathbb{S}^{n-1}\rightarrow\mathbb{R}} \), any \( {y\in\mathbb{S}^{n-1}} \), and any spherical harmonics \( {Y_k} \) on \( {\mathbb{S}^{n-1}} \) of degree \( {k\geq0} \),
\[ \int f(x\cdot y)Y_k(x)\sigma_{\mathbb{S}^{n-1}}(\mathrm{d}x) =\lambda_kY_k(y) \]
where
\[ \lambda_k=\frac{\Gamma(\frac{n}{2})}{\sqrt{\pi}\Gamma(\frac{n-1}{2})}\int_{-1}^1f(t)P_k(t)(1-t^2)^{\frac{n-3}{2}}\mathrm{d}t. \]
We recover the simple Funk-Hecke formula by taking \( {k=0} \) for which \( {Y_0=1} \) and \( {P_0=1} \).
In other words, for any ``kernel'' \( {x\mapsto f(x\cdot y)} \) on \( {\mathbb{S}^{n-1}} \), a spherical harmonics \( {Y_k} \) of degree \( {k} \) is an eigenfunction, associated to the eigenvalue \( {\lambda_k} \), of the Funk transform
\[ g\mapsto \int f(x\cdot\bullet)g(x)\sigma_{\mathbb{S}^{n-1}}(\mathrm{d}x). \]
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)