Canonical Bessel transmutation operators with applications to wavelet admissibility

Motivated by recent advances in canonical harmonic analysis for Sturm–Liouville transforms due to Soltani and collaborators, including Parseval–Goldstein type identities and Boas-type results for canonical Sturm–Liouville settings, we develop an Integral Transforms Composition Method (ITCM) for the canonical Bessel operator Δνm associated with the Ghazouani-Sahbani transform. By linking the classical Fourier-Bessel transform Fν to its canonical analogue Fνm via spectral unitary dilation, we establish a sharp spectral alignment identity. We construct unitary transmutation operators Sm and Pm that intertwine Δνm with the classical Bessel operator Bν. Using Weber-Sonine formulae, we derive closed-form kernels involving Bessel functions and quadratic phase factors. We apply this framework to the transport of wavelet admissibility from classical to canonical settings. These results extend the ITCM framework to Bessel operators with quadratic phase deformations, bridging classical and canonical harmonic analysis.