What the study found
The authors show an analogue of the classical Paneah-Logvinenko-Sereda theorem for the Fourier-Bessel transform. In this setting, if a set E in the positive real numbers is relatively dense with respect to the measure μ_α, then the L2 norm of a function can be controlled by its L2 norm on E, provided the Fourier-Bessel transform of the function is supported in [R, R+1].
Why the authors say this matters
The abstract says the work is motivated by control theory questions about decay rates for the damped wave equation. The authors present the result as an uncertainty principle for the Fourier-Bessel transform, which is the stated mathematical framework for that motivation.
What the researchers tested
They consider sets E ⊂ R+ that are μ_α-relatively dense, where dμ_α(x) is comparable to x^(2α+1) dx and α > -1/2. They study functions f in L2_α(R+) whose Fourier-Bessel transform F_α(f) is supported in [R, R+1].
What worked and what didn't
They prove that under those conditions, ||f||_{L2_α(R+)} is bounded by a constant times ||f||_{L2_α(E)}. The abstract does not describe any negative result or case where the bound fails.
What to keep in mind
The available summary does not give the proof, constants, or further limitations beyond the stated assumptions on α, E, and spectral support. It also does not describe applications beyond the motivation from control theory.
Key points
- The paper gives a Fourier-Bessel analogue of the Paneah-Logvinenko-Sereda theorem.
- If E is μ_α-relatively dense and the Fourier-Bessel transform of f is supported in [R, R+1], then ||f|| can be controlled by ||f|| on E.
- The measure μ_α is described as dμ_α(x) ≈ x^(2α+1) dx, with α > -1/2.
- The abstract says the work is motivated by control theory and decay rates for the damped wave equation.
- No failures, counterexamples, or detailed limitations are described in the abstract.
- The result is framed as an uncertainty principle for the Fourier-Bessel transform.
Disclosure
- Research title:
- Fourier-Bessel uncertainty result for relatively dense sets
- Authors:
- Benjamin Jaye, Rahul Sethi
- Institutions:
- Georgia Institute of Technology
- Publication date:
- 2026-04-20
- OpenAlex record:
- View
Get the weekly research newsletter
Stay current with peer-reviewed research without reading academic papers — one filtered digest, every Friday.