A failure in the transfer test
The paper gives a counterexample to a transfer question: it constructs a smooth radial function that is strictly positive definite in Euclidean space but not positive definite after direct geodesic substitution on a sphere. That construction covers every even dimension d = 2m with m ≥ 1 and every prescribed support radius R in (0, π].
The underlying question is whether Euclidean spectral positivity forces spherical spectral positivity when the support of the radial function lies within [0, π]. In ordinary terms, the paper tests whether passing the Euclidean positivity condition is enough to pass the corresponding spherical condition; its construction shows that it is not in the even-dimensional cases covered here.
Building the exact-support example
The authors begin with a nonzero, real, smooth radial function q whose compact support is exactly the d-dimensional ball of radius R/2. They convolve q with itself and evaluate the result along the first coordinate direction to form the candidate radial function φ.
The resulting radial extension is smooth and compactly supported, with support radius exactly R. The Titchmarsh–Lions support theorem supplies the calculation: two support balls of radius R/2 add to a ball of radius R.
A spectral sign flips the result
On Euclidean space, the kernel φ(∥x − y∥₂) is strictly positive definite. On the sphere, its degree-two Gegenbauer coefficient is negative, and Schoenberg’s characterization treats that negative coefficient as a certificate that the spherical kernel is not positive definite.
The proof turns this spherical coefficient into a Fourier question. A lemma represents it as a Fourier pairing involving a radial window W_d and the squared magnitude of the Fourier transform of q. The window is integrable and supported in a ball of radius d + 1, while it is negative at all sufficiently small nonzero frequencies.
To use that low-frequency sign, the authors apply a high power of the polynomial Laplacian filter I + Δ/(d + 1)^2. The filter concentrates the relevant Fourier energy near the origin without enlarging the function’s physical support.
For every even d ≥ 2, a dimension-dependent threshold M_d makes the filtered energy negative for every 0 < R ≤ π/2 and every M ≥ M_d. Scaling by R/2 and taking the self-convolution then carries this smaller-radius result to the final construction with exact support throughout 0 < R ≤ π.
What the parity result means
Together, these steps show that direct transfer cannot be guaranteed in even dimensions: each allowed even dimension and support radius has at least one smooth, strictly Euclidean-positive counterexample. They do not say that every Euclidean-positive function fails on a sphere.
The broader parity conclusion combines this construction with a cited odd-dimensional result. The paper states that the direct transfer property holds if and only if d is odd for every integer d ≥ 2, while the odd-dimensional implication is attributed to prior work rather than developed in the supplied construction.
Preprint status and disclosure
The work is an arXiv preprint, version 2 in the math.CA category, dated 27 August 2026.
The authors say the work was supported in part by China’s National Natural Science Foundation through grants 12671642 and 12371103, and by the Guangdong Basic and Applied Basic Research Foundation through grant 2024A1515011194.
They also disclose using ChatGPT and Codex for English editing, preliminary numerical exploration, auxiliary computations and integral estimates, and locating standard integral identities. They state that they independently verified the theorem, construction and proofs.
Paper data and sources
Original title: Exact-Support Counterexamples to Euclidean-to-Spherical Transfer of Positive Definiteness in Even Dimensions
Authors: Wentao Huang, Haizhang Zhang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text