LAUSR

This paper deals firstly with some q-harmonic analysis properties for the q-windowed Bessel Fourier transform related to the q-Bessel function of the third kind as Plancherel formula, inversion formula in $$\mathcal {L}_{q,2,\nu }$$Lq,2,ν. Secondly, we give a weak uncertainty principle for it and we show that the portion of the q-windowed Bessel Fourier transform lying outside some set of finite measure cannot be arbitrarily too small. Then, we verify a version of Heisenberg–Pauli–Weyl type uncertainty inequalities for the q-windowed Bessel Fourier transform and its generalization. Finally, using the kernel reproducing theory, given by Saitoh (Theory of reproducing kernels and its applications. Longman Scientific and Technical, Harlow, 1988), we will be able to realize the natural and powerful approximation problems that lead to the q-windowed Bessel Fourier transform inverses.