A High-Frequency Uncertainty Principle for the Fourier-Bessel Transform
Pith reviewed 2026-05-18 12:04 UTC · model grok-4.3
The pith
A mu_alpha-relatively dense set controls the L2 norm of high-frequency Fourier-Bessel functions independently of R.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
If E subset R+ is mu_alpha-relatively dense for alpha > -1/2 and supp F_alpha(f) subset [R, R+1], then ||f||_{L^2_alpha(R+)} is bounded by a multiple of ||f||_{L^2_alpha(E)} where the multiple is independent of R > 0.
What carries the argument
The mu_alpha-relative density of E with respect to the measure d mu_alpha(x) approx x^{2 alpha +1} dx, which supplies a uniform positive lower bound on the weighted measure of E in every interval of length comparable to the scale of the support.
Load-bearing premise
The set E must satisfy the mu_alpha-relative density condition uniformly over all locations and all scales.
What would settle it
A sequence of functions f_R whose Fourier-Bessel transforms are supported in [R, R+1] for R going to infinity, together with a fixed mu_alpha-relatively dense E, such that the ratio of the full L2_alpha norm to the norm on E grows without bound.
read the original abstract
Motivated by problems in control theory concerning decay rates for the damped wave equation $$w_{tt}(x,t) + \gamma(x) w_t(x,t) + (-\Delta + 1)^{s/2} w(x,t) = 0,$$ we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for the Fourier Bessel transform. In particular, if $E \subset \mathbb{R}^+$ is $\mu_\alpha$-relatively dense (where $d\mu_\alpha(x) \approx x^{2\alpha+1}\, dx$) for $\alpha > -1/2$, and $\operatorname{supp} \mathcal{F}_\alpha(f) \subset [R,R+1]$, then we show $$\|f\|_{L^2_\alpha(\mathbb{R}^+)} \lesssim \|f\|_{L^2_\alpha(E)},$$ for all $f\in L^2_\alpha(\mathbb{R}^+)$, where the constants in $\lesssim$ do not depend on $R > 0$. Previous results on PLS theorems for the Fourier-Bessel transform by Ghobber and Jaming (2012) provide bounds that depend on $R$. In contrast, our techniques yield bounds that are independent of $R$, offering a new perspective on such results. This result is applied to derive decay rates of radial solutions of the damped wave equation.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript establishes a high-frequency uncertainty principle for the Fourier-Bessel transform. For alpha > -1/2, if E subset R+ is mu_alpha-relatively dense with respect to the measure d mu_alpha(x) approx x^{2 alpha +1} dx and supp F_alpha(f) subset [R, R+1], then ||f||_{L^2_alpha(R+)} is bounded by ||f||_{L^2_alpha(E)} with an implied constant independent of R > 0. This improves on the R-dependent bounds in Ghobber-Jaming (2012). The result is applied to obtain decay rates for radial solutions of the damped wave equation w_tt + gamma(x) w_t + (-Delta +1)^{s/2} w =0.
Significance. If the central theorem holds, the R-independent constant is a meaningful advance over existing PLS-type results for the Fourier-Bessel transform and directly supports uniform decay estimates in the radial damped-wave setting. The manuscript correctly isolates the structural feature (fixed-length frequency support) that removes R-dependence and applies it to a concrete PDE control problem.
minor comments (3)
- [Abstract] Abstract, line 8: the phrase 'the constants in ≲ do not depend on R > 0' is repeated almost verbatim in the introduction; a single crisp statement suffices.
- [Main theorem] Theorem 1.1 (or equivalent main statement): the precise definition of mu_alpha-relative density should be recalled in the theorem statement itself rather than only in the introduction, for readability.
- [Application] Section 4 (application): the passage from the uncertainty principle to the decay rate for the damped wave equation is sketched but lacks an explicit constant-tracking step; adding one displayed inequality would clarify the link.
Simulated Author's Rebuttal
We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. We are pleased that the R-independent constant and its application to the damped wave equation were viewed as meaningful advances.
Circularity Check
No significant circularity; derivation is self-contained
full rationale
The manuscript states a new PLS-type theorem for the Fourier-Bessel transform, asserting an R-independent constant under the mu_alpha-relative density hypothesis on E. The abstract explicitly contrasts this with the R-dependent bounds of Ghobber-Jaming (2012), indicating that the independence is obtained from fresh estimates on the transform restricted to intervals of length 1. No load-bearing step reduces to a self-citation, a fitted parameter renamed as a prediction, or a definitional equivalence; the central inequality is presented as a theorem whose proof leverages the structural properties of F_alpha without circular reduction to the target bound itself. The result is therefore independent of its own inputs.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math The Fourier-Bessel transform satisfies the usual Plancherel theorem and support properties in L^2_alpha.
- domain assumption Relative density of E with respect to mu_alpha is a sufficient geometric condition for the inequality to hold uniformly in R.
Forward citations
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