UNCERTAINTY PRINCIPLES FOR THE GENERALIZED LINEAR CANONICAL FOURIER-BESSEL TRANSFORM

We introduce a generalized differential operator \(\varvec{\Delta }^{\varvec{m}}_{\varvec{\alpha },\varvec{n}}\) ${\mathbf{\Delta }}_{\mathit{\alpha },\mathit{n}}^{\mathit{m}}$ , which extends the functionality of the Bessel type operator \(\varvec{\Delta }^{\varvec{m}}_{\varvec{\alpha }}\) ${\mathbf{\Delta }}_{\mathit{\alpha }}^{\mathit{m}}$ . The aim of this paper is to develop a new harmonic analysis related to \(\varvec{\Delta }^{\varvec{m}}_{\varvec{\alpha },\varvec{n}}\) ${\mathbf{\Delta }}_{\mathit{\alpha },\mathit{n}}^{\mathit{m}}$ . We define the generalized canonical Fourier-Bessel transform \(\varvec{\mathcal {F}}^{\varvec{m}}_{\varvec{\alpha },\varvec{n}}\) ${\mathcal{F}}_{\mathit{\alpha },\mathit{n}}^{\mathit{m}}$ , and study some of its important properties. Some useful properties of the considered transform such as Riemann-Lebesgue lemma, inversion formula, operational formulas, Plancherel formula, Paley-Wiener theorem, and Babenko type inequality are derived. In the present paper the Heisenberg inequality, Hardy theorem, Nash-type inequality, Carlson-type inequality, and global uncertainty principle are given.