A mathematical preprint has found that a known differentiability lower bound for isotropic positive definite functions is optimal on every even-dimensional sphere. In other words, the guarantee cannot generally be raised by one more derivative. The paper supplies explicit strictly positive definite examples that reach the guaranteed interior order and fail at the next one, with the failure appearing at the equator.
This is a result about mathematical objects rather than a data set. The work starts with a seed on S2 and a sparse Schoenberg sequence, then builds functions on higher-dimensional spheres. Here, differentiability is simply the number of derivatives a function has in a region, so the question is where that smoothness must stop. No empirical sample is involved.
A theorem meets its boundary
The benchmark is a lower-bound theorem. For dimension d >= 1 and k >= 0, it says that when a function's even continuation at zero is 2k times differentiable, the function has derivatives through order 2k + floor((d - 1)/2) throughout the interior interval from 0 to pi. In plain language, smoothness at the endpoint guarantees a prescribed amount of smoothness inside the sphere. The central issue was whether this interior guarantee was merely conservative in even dimensions.
The new construction hits that count exactly. Write an even dimension as d = 2m, with m >= 1. For every k >= 0, its function has the required 2k derivatives at zero and derivatives through order 2k + m - 1 in the interior. But the derivative of order 2k + m does not exist at the fixed equatorial point pi/2. That one-step failure is what makes the lower bound sharp: the theorem guarantees the derivatives up to one order, while the example blocks the next.
The construction starts in two dimensions
The proof begins in two dimensions. Its seed function belongs to Ψ2, but its derivative at the equator pi/2 does not exist. The construction starts from a sparse Schoenberg coefficient sequence and uses it as the base for functions on higher spheres.
The paper then makes the seed strictly positive definite without repairing the defect. After the perturbation, the function lies in Ψ+2, every Schoenberg coefficient is positive, and the derivative at pi/2 is still absent. Strict positive definiteness and equatorial nondifferentiability therefore coexist in the constructed seed.
To carry the example into higher even dimensions, the proof uses turning bands and spherical montée, the dimension-walk operations described in the paper. The missing equatorial derivative remains absent after each operation. The spherical montée lemma also states that if the even continuation of a function has 2q derivatives at zero, the even continuation after the operation has 2q + 2 derivatives there.
A precise edge, not an empirical performance claim
The proof is organized around Schoenberg expansions, a moment criterion at the origin, and dimension-walk formulas. To establish the missing equatorial derivative, a Poisson-boundary argument examines regularized derivatives: they would have to converge to a finite derivative of the boundary function, which contradicts the paper's established nonexistence of that derivative.
The construction also marks the pole limit. Although the even continuation has 2k derivatives at zero, it is not 2k + 2 times differentiable there. The examples are also not positive definite in the next dimension. Together, these results show a boundary both in how much smoothness can be guaranteed and in how far the same positivity construction can be carried across dimensions.
The paper's final corollary summarizes the optimal index by parity: in even dimension d = 2m, the guaranteed order is 2k + m - 1 and the first non-guaranteed derivative is 2k + m. The document is an arXiv preprint, version v1, dated 26 August 2026. Its conclusions concern the defined class of isotropic positive definite functions on spheres and rest on constructed examples and proofs, not empirical performance measurements.
Paper data and sources
Original title: Optimal differentiability of isotropic positive definite functions on even-dimensional spheres
Authors: Yan Ge
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text