Abstract
We study the class of q-Fourier multiplier operators T-m := F-q (mF(q)), which are acted on the q-Sobolev space H-*,q(S) (R-q), and we obtain the exact expression and some properties for the extremal functions of the best approximation problem in quantum calculus inf f is an element of H-*,q(S) (R-q){eta parallel to f parallel to(2)(H*,qs) (R-q) + parallel to g - T(m)f parallel to(2)(L2(Rq,+))}, where eta > 0 and g is an element of L-2(R-q,+). As an application, we provide numerical approximate formulas for a limit case eta up arrow 0; using q-calculus, which generalizes the Gauss-Kronrod method studied given in [14] in one-dimensional space.