Prove the proposed explicit spectral bound for digital-line Gram matrices

Prove that the explicit bound Λ_k=min{N^2,2N(1+log N)/|ω_k|} majorizes the largest eigenvalue λ_max(M_k) of every offset-frequency Gram matrix M_k=A_k^*A_k associated with the cyclic single-quadrant approximate discrete Radon transform.

Background

The polynomial-filter realization requires upper bounds Λ_k≥λ_max(M_k) to control the Chebyshev approximation intervals. For exact straight lines, a Dirichlet-kernel argument yields a bound of the proposed form.

For integer-sampled digital lines, the multilevel phase increments do not double in the manner required by the straight-line proof. Numerical experiments indicate that the proposed bound remains above the actual largest eigenvalue, despite failing to dominate the largest row sums. Establishing the bound therefore requires exploiting cancellation in the eigenvalues rather than relying solely on row-sum estimates.

References

We leave this open. Nothing about exactness rests on it in practice: by Theorem~\ref{thm:fast}(i) the identity $Ln_pRn=I$ needs only an invertible $E\top F_pE$, which the bound $\Lambda_k\ge\lambda_{\max}(M_k)$ guarantees and which holds in all computed cases; what the bound controls is the accuracy of the filters.

— Explicit Oblique Inverses of the Approximate Discrete Radon Transform  (2610.23363 - Rim, 16 Oct 2026) in Section 5, subsection “An explicit choice of Λ_k”