Analytical approximation formulas in quantum calculus

We study the class of q-Fourier multiplier operators T m : = F q ( F 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 ∈ H * , q s ( ℝ q ) { η ∥ f ∥ H * , q s ( ℝ q ) 2 + ∥ g − T m f ∥ L 2 ( ℝ q , + ) 2 } , where η > 0 and g ∈ L 2 ( ℝ q , + ) . As an application, we provide numerical approximate formulas for a limit case η ↑ 0 ; using q-calculus, which generalizes the Gauss-Kronrod method studied given in [14] in one-dimensional space.