Best approximate inversion formulas for Quantum Bessel multiplier operators

Using the \(q\)-Jackson integral and some elements of the \(q\)-harmonic analysis associated with the generalized q-Bessel operator for fixed \(0<q<1\), we introduce the generalized q-Bessel multiplier operators and we give some new results related to these operators as Plancherel’s, Calderón's reproducing formulas and Heisenberg's, Donoho-Stark's uncertainty principles. Next, using the theory of reproducing kernels we give best estimates and an integral representation of the extremal functions related to these operators on weighted Sobolev spaces.