A High-Frequency Uncertainty Principle for the Fourier-Bessel Transform
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| Format: | Preprint |
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2025
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| author | Jaye, Benjamin Sethi, Rahul |
| author_facet | Jaye, Benjamin Sethi, Rahul |
| contents | Motivated by problems in control theory concerning decay rates for the damped wave equation $$w_{tt}(x,t) + γ(x) w_t(x,t) + (-Δ+ 1)^{s/2} w(x,t) = 0,$$ we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for the Fourier Bessel transform. In particular, if $E \subset \mathbb{R}^+$ is $μ_α$-relatively dense (where $dμ_α(x) \approx x^{2α+1}\, dx$) for $α> -1/2$, and $\operatorname{supp} \mathcal{F}_α(f) \subset [R,R+1]$, then we show $$\|f\|_{L^2_α(\mathbb{R}^+)} \lesssim \|f\|_{L^2_α(E)},$$ for all $f\in L^2_α(\mathbb{R}^+)$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_25500 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A High-Frequency Uncertainty Principle for the Fourier-Bessel Transform Jaye, Benjamin Sethi, Rahul Classical Analysis and ODEs Motivated by problems in control theory concerning decay rates for the damped wave equation $$w_{tt}(x,t) + γ(x) w_t(x,t) + (-Δ+ 1)^{s/2} w(x,t) = 0,$$ we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for the Fourier Bessel transform. In particular, if $E \subset \mathbb{R}^+$ is $μ_α$-relatively dense (where $dμ_α(x) \approx x^{2α+1}\, dx$) for $α> -1/2$, and $\operatorname{supp} \mathcal{F}_α(f) \subset [R,R+1]$, then we show $$\|f\|_{L^2_α(\mathbb{R}^+)} \lesssim \|f\|_{L^2_α(E)},$$ for all $f\in L^2_α(\mathbb{R}^+)$, 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. |
| title | A High-Frequency Uncertainty Principle for the Fourier-Bessel Transform |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2509.25500 |