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Analytical approximation formulas in quantum calculus. (English) Zbl 1395.42006

Summary: We study the class of \(q\)-Fourier multiplier operators \(T_m:=\mathcal F_q(m\mathcal F_q)\), which are acted on the \(q\)-Sobolev space \(\mathcal H^s_{\ast,q},(\mathbb{R}_q)\), and we obtain the exact expression and some properties for the extremal functions of the best approximation problem in quantum calculus \(\inf_{f\in\mathcal{H}^{s}_{\ast,q}(\mathbb{R}_{q})}\{\eta\|f\|^2_{\mathcal{H}^{s}_{\ast,q}(\mathbb{R}_{q})}+\|g-T_mf\|^2_{L^{2}(\mathbb{R}_{q,+})}\}\), where \(\eta>0\) and \(g\in L^2(\mathbb{R}_{q,+})\). 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.

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
41A50 Best approximation, Chebyshev systems
42A45 Multipliers in one variable harmonic analysis
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