We study the class of q-Fourier multiplier operators Tm:=Fq(Fq), which are acted on the q-Sobolev space H*,qs(Rq), and we obtain the exact expression and some properties for the extremal functions of the best approximation problem in quantum calculus inffH*,qs(Rq){||f||H*,qs(Rq)2+||g–Tmf||L2(Rq,+)2}, where > 0 and gL2(Rq,+). As an application, we provide numerical approximate formulas for a limit case 0; using q-calculus, which generalizes the Gauss-Kronrod method studied given in [14] in one-dimensional space.