Quantitative Uncertainty Principles for the Canonical Fourier—Bessel Transform

Prolate spheroidal wave functions associated with the canonical Fourier–Bessel transform and uncertainty principles

“…In the present work, we continue the analysis begun in previous literature [2,6] to understand more the canonical Fourier-Bessel transform. We will here concentrate to establish the finite canonical Fourier-Bessel transform operator…”

“…In the present work, we continue the analysis begun in previous literature [2, 6] to understand more the canonical Fourier–Bessel transform. We will here concentrate to establish the finite canonical Fourier–Bessel transform operator $$$ {\mathcal{F}}_{\nu, r}^{\mathbf{m}} $$$ on $$$ {\mathcal{L}}_{2,\nu}\left(\left[0,1\right]\right) $$$, the self‐adjoint compact operator $$$ {\mathcal{A}}_{\nu, r}^{\mathbf{m}}={\mathcal{F}}_{\nu, r}^{{\mathbf{m}}^{-\mathbf{1}}}\kern3pt \circ \kern3pt {\mathcal{F}}_{\nu, r}^{\mathbf{m}} $$$ and the corresponding eigenfunctions $$$ {\left\{{\varphi}_{n,r}^{\left(\nu \right)}\right\}}_{n=0}^{\infty } $$$ on $$$ P{B}_{\nu, r}^{\mathbf{m}} $$$, and the Paley–Wiener space of canonical Bessel band‐limited functions.…”

“…In earlier studies [2,6], gigantic investigations and efforts have been provided by the authors to study the canonical Fourier-Bessel transform by consideration of its broad field of applications, especially that it encompasses several integral transformations, and several of its important properties are given, including Riemann-Lebesgue lemma, inversion formula and operational formulas, Plancherel theorem and Babenko-Beckner inequality, and uncertainty principles.…”