Analytical approximation formulas in quantum calculus

Abstract

We study the class of q-Fourier multiplier operators $T_m:={\mathcal{F}}_q({\mathcal{F}}_q)$ , which are acted on the q-Sobolev space ${\mathcal{H}}_{*,q}^s({\mathrm{ℝ}}_q)$ , and we obtain the exact expression and some properties for the extremal functions of the best approximation problem in quantum calculus $\underset{f\in {\mathcal{H}}_{*,q}^s({\mathrm{ℝ}}_q)}{inf}\{\eta \parallel f{\parallel }_{{\mathcal{H}}_{*,q}^s({\mathrm{ℝ}}_q)}^2+\parallel g-T_{m}f{\parallel }_{L^2({\mathrm{ℝ}}_{q,+})}^2\}$ , where $\eta >0$ and $g\in L^2({\mathrm{ℝ}}_{q,+})$ . As an application, we provide numerical approximate formulas for a limit case $\eta \uparrow 0$ ; using q-calculus, which generalizes the Gauss-Kronrod method studied given in [14] in one-dimensional space.