Establish a logarithmic lower bound for the Tikhonov frame operator

Prove that, for a fixed positive ratio τ/N, the smallest eigenvalue of the window-compressed Tikhonov frame operator E^T F_τ E satisfies λ_min(E^T F_τ E)≳1/log N as N tends to infinity.

Background

The outer Chebyshev polynomial used to invert the frame operator requires a reliable lower bound on λmin(ET Fτ E). Numerical values for τ=0.2N are compatible, over the tested range, with a 1/log N lower bound, but the margin decreases as N grows.

A proof of this estimate would imply that the outer polynomial degree is O(√log N) with explicit constants. The paper cautions that the empirical trend may eventually fall below the candidate 4/log N bound, motivating the weaker unresolved question stated here.

References

Whether $\lambda_{\min}(E\top F_\tau E)\gtrsim1/\log N$ holds for fixed $\tau/N$ is open; a proof would make the outer degree $O(\sqrt{\log N})$ with explicit constants.

— Explicit Oblique Inverses of the Approximate Discrete Radon Transform  (2610.23363 - Rim, 16 Oct 2026) in Section 5, subsection “The lower end of the outer interval”