What the study found
The study shows a Fourier-Bessel transform analogue of the classical Paneah-Logvinenko-Sereda theorem. For functions whose Fourier-Bessel transform is supported in a frequency interval [R, R+1], the authors prove an L2 bound by the function's size on a relatively dense set.
Why the authors say this matters
The abstract says the work is motivated by control theory problems, especially decay rates for the damped wave equation. The authors present the result as relevant to that setting by giving a frequency-localized estimate in the Fourier-Bessel framework.
What the researchers tested
They studied functions on the positive real line with respect to the measure dμ_α(x) ≈ x^{2α+1} dx, for α > -1/2. They assumed the set E ⊂ R+ is μ_α-relatively dense and that supp F_α(f) ⊂ [R, R+1], where F_α denotes the Fourier-Bessel transform.
What worked and what didn't
Under those assumptions, they prove that the L2_α norm of f on the whole space is controlled by its L2_α norm on E, written as ‖f‖_{L^2_α(R+)} ≲ ‖f‖_{L^2_α(E)}. The abstract does not report cases where the estimate fails or describe negative results.
What to keep in mind
The abstract gives only the stated theorem and its setup, not proof details. It also does not describe limitations beyond the conditions already listed: α > -1/2, relative density of E, and frequency support contained in [R, R+1].
Key points
- The paper proves a Fourier-Bessel version of the Paneah-Logvinenko-Sereda theorem.
- The result applies when the Fourier-Bessel transform is supported in a unit-length interval [R, R+1].
- A μ_α-relatively dense set E controls the L2_α norm of the function on the whole positive real line.
- The work is motivated by control theory questions for the damped wave equation.
- The abstract does not describe failures, exceptions, or proof details.
Disclosure
- Research title:
- High-frequency uncertainty principle for Fourier-Bessel transform
- Authors:
- Benjamin Jaye, Rahul Sethi
- Institutions:
- Georgia Institute of Technology, Georgia Institute of Technology
- Publication date:
- 2026-04-20
- OpenAlex record:
- View
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