Extra rounds of a numerical procedure called Sloan iteration do not guarantee its ideal convergence gain once a spherical integral-equation calculation is discretized, according to an arXiv preprint. The analysis separates the error into a filtered spectral tail and an aliasing term—a discretization contribution that can remain when the tail is reduced.
The ideal bounds
The work covers zonal integral equations on the sphere and Galerkin and degenerate-kernel approximations, along with discrete Galerkin and product-integration Nyström versions. In the quadrature-free case, the Galerkin uniform-norm error—the maximum error over the sphere—is bounded at order n^{−(r+qβ−1)}. The degenerate-kernel/Nyström version is bounded at order n^{−(r+(q+1)β−1)} under the stated multiplier assumptions, adding one multiplier-smoothing factor relative to Galerkin.
When the grid enters the picture
With polynomially exact quadrature, the discrete error is bounded by the sum of a filtered spectral-tail term and a filtered aliasing term. The tail term carries the n-dependent part of the bound, while the aliasing term carries the s-dependent part. Under the paper’s exact multiplier-order, nondegeneracy, positivity, exactness and cardinality assumptions, the two branches have a sharp balance scale, written as s^crit_{n,⋆}=n^{(r+γ⋆−1)/r}.
Sampling can preserve the split
When sampling is inexact, the paper replaces hyperinterpolation with a weighted least-squares projector corrected by the Gram matrix. The corrected estimate retains the tail–aliasing decomposition and makes the aliasing contribution depend explicitly on the Marcinkiewicz–Zygmund, or MZ, stability constant.
The sharp inexact-sampling branch is conditional. If γ⋆>1, the constant mode is active (μ0≠0), sampling deviations remain uniformly below 1, and the number of nodes is at most proportional to n², MZ-stable sampling retains a saturation branch—the sampling-limited rate of order n^−r. MZ stability alone does not imply recovery of the quadrature-free rate.
The numerical checks
The numerical model set r=3 and used multipliers μℓ=0.55(1+ℓ)^−3/2, with |1−μℓ| bounded below by 0.45. Rates were estimated as negative least-squares slopes over the five largest n values. The exact-quadrature experiment tested q=0 and q=1 at t=2n and s=n, then tested q=1 at the recovery scale s∼n^{5/3}.
At the minimum exactness tested, one Sloan step changed the observed spectral-tail slope from 3.32 to 4.72, while the separate certificate rate remained 2.97. At the recovery scale, the tail and certificate rates were 4.52 and 4.38.
In the inexact point-family tests, the largest listed sampling-deviation value was 0.0719 and the largest Gram-matrix condition number was 1.142. Total-error slopes ranged from 3.84 to 4.46, compared with spectral-tail reference slopes of 4.67 for q=1 and 6.03 for q=2.
A conditional mathematical result
The result is a worst-case mathematical statement for zonal operators on the sphere. Its sharp two-branch conclusion is conditional on the stated multiplier order, nondegeneracy, positivity, exactness and cardinality assumptions. The numerical slopes came from finite deterministic configurations, with no uncertainty intervals or replicate-based variability reported.
For numerical analysts choosing a discretization, the central design trade-off is resolution: extra Sloan depth may require a matching increase in quadrature resolution if the spectral-tail gain is to survive.
Publication status
The document is an arXiv preprint, version 1, dated 20 Aug 2026.
Paper data and sources
Original title: Superconvergence and aliasing saturation in Sloan iteration for spherical integral equations
Authors: Hao-Ning Wu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text