Unverified50% confidenceFactExact time
On the unit sphere, if the kernel of an integral operator depends only on the inner product of two unit vectors, then the eigenfunctions of the operator are exactly the spherical harmonics, and the eigenvalues can be given explicitly by the Legendre expansion coefficients of the kernel.
1
Sources
50%
Confidence
Long-term
Relevance
10/27/2026
First Seen
Sources
Related Entities
Related Claims
UnverifiedSpherical harmonics characterize angular momentum by order l, where l=0 corresponds to scalars, l=1 corresponds to vectors, and higher orders correspond to tensor representations.72% similarUnverifiedA complex number is essentially a two-dimensional object, naturally corresponding to a point or vector on the plane.71% similarUnverifiedFor harmonic functions satisfying the Laplace equation, the value at any interior point equals the average of the function values on the sphere centered at that point—a property known as the mean value property.70% similarUnverifiedThe squared exponential kernel corresponds to the class of infinitely smooth functions, while the Matérn-ν kernel corresponds to the class of ν-times differentiable functions, with smaller ν yielding rougher function classes.68% similarUnverifiedAccording to Mercer's theorem, any kernel function satisfying the positive definiteness condition corresponds to an inner product in some feature space, which allows nonlinear problems to be transformed into linear problems in a high-dimensional space.67% similar
Cite This Claim
Stable URI
https://kongchang.com/claim/1102476API
curl https://kongchang.com/api/v1/knowledge/claims/1102476MCP
get_claim(id=1102476)