bibkey: karnikrombergdavenport2021prolate authors: Santhosh Karnik, Justin Romberg, Mark A. Davenport year: 2021 title: Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences doi: 10.1016/j.acha.2021.04.002 url: https://arxiv.org/abs/2006.00427v2 claim: Explicit prolate concentration eigenvalue and trace-tail bounds give effective finite-rank errors for that operator. The physical prolate and additive Weil-band parameters differ, and neither bound supplies an actual-Weil finite-section error or arithmetic positivity. strata_touched: [] license: citation-only triage: anchor

Effective concentration bounds and the operator interface

The primary text inspected is arXiv:2006.00427v2, associated with Applied and Computational Harmonic Analysis 55 (2021), 97–128, DOI 10.1016/j.acha.2021.04.002. The 29-page PDF SHA-256 is 7c56255db5bd6454e3fc926952052fe466ffd49c4961ad8206638dcac1dd39a0. Section 2.2 and Corollaries 3–4, printed pp.3–4 and p.9, specify the operator, Fourier convention, constants and index ranges. The bounds are reused as source results; their proofs and numerical experiments are not reproduced here.

Explicit source bounds

Write $\gamma$ for the source’s concentration parameter, reserving $c$ for the project’s prime-power cutoff. On $L^2([-1,1])$, use the sinc concentration operator

$$(S_{\gamma }h)(x)={\int }_{-1}^1\frac{\sin (\gamma (x-y))}{\pi (x-y)}h(y)dy.$$

Its decreasing eigenvalues are indexed from zero. Define

$$b_{\gamma }=\frac{2}{{\pi }^2}\log (100\gamma /\pi +25),m_{\gamma }=\lceil 2\gamma /\pi \rceil .$$

The upper half of Corollary 3 states

$${\mu }_k(\gamma )\le 10\exp (-(k-m_{\gamma }-6)/b_{\gamma }),k\ge m_{\gamma }.$$

Corollary 4 states

$$\sum _{k=K}^{\infty }{\mu }_k(\gamma )\le 10b_{\gamma }\exp (-(K-m_{\gamma }-7)/b_{\gamma }),K\ge m_{\gamma }.$$

These are effective bounds with explicit constants, rather than fixed-mode asymptotics. Retaining modes $0,\dots ,K-1$ of this same operator gives an operator-norm error bounded by the first omitted eigenvalue and a trace-norm error bounded by the displayed sum. In particular, for $0<\epsilon <1$, the source’s eigenvalue bound supplies an operator-norm allowance $\epsilon$ when the integer $K$ satisfies

$$K\ge m_{\gamma }+6+b_{\gamma }\log (10/\epsilon ).$$

This spectral truncation uses concentration eigenmodes. It is not a statement about an arbitrary retained polynomial or trigonometric basis.

Two parameter maps that must remain distinct

In the source’s angular-frequency convention $e^{-i\omega t}$, a physical interval $[-T/2,T/2]$ and band $[-\Omega ,\Omega ]$ have $\gamma =\Omega T/2$.

For the first construction the finite-rank allowance above reads

$$K\ge \lceil 4c\rceil +6+\frac{2}{{\pi }^2}\log (200c+25)\log (10/\epsilon ).$$

For the second construction, substitute $\gamma =\Omega L$ in the source bound instead. Neither substitution identifies the two physical spaces or their retained bases.

What still needs an actual-Weil estimate

The 2026 spectral source already supplies the auxiliary transform convergence. Applying the present concentration bounds to that auxiliary operator does not bound the difference from its actual Weil ground eigenfunction. That needs a quantitative arithmetic transport, a common norm and the specified normalization.

For the existing retained Weil comparison, the modes are even Legendre polynomials in the additive window. An error bound in concentration eigenmodes cannot be assigned to that projection without a justified basis comparison. More generally, concentration-operator norm control does not pay for the actual Gamma, prime translations, poles or their couplings to the discarded test space. The unresolved finite-section allowance $r(L,N)$ concerns the full common form, not $S_{\gamma }$.

No new eigenvalue experiment, actual-Weil lower bound, kernel verification or RH proof is provided by this source note. Its reusable contribution is the explicit concentration error supplier with its correct operator and parameters.