On the Jackson-type inequalities in approximation theory connected to the \(q\)-Dunkl operators in the weighted space \(L^2_{q, \alpha }(\mathbb{R}_q, |x|^{2 \alpha +1}d_q x )\)
Let \(L^2_{q,\alpha} (\mathbb {R}_q)\) be the space of functions with the norm \[ \|f\|_{2,\alpha ,q}=\{ (\int _{-\infty}^\infty |f(x)|^2|x|^{2\alpha +1}d_qx)^{1/2}<\infty \}, \] where \(0<q<1,\ \alpha \leq -1/2\) and the integral is understood in the sense of \(q\)-definite Jackson integral. In the work of \textit{N. Bettaibi} and \textit{R. Bettaieb} [Tamsui Oxf. J. Math. Sci. 25, No. 2, 177--205 (2009; Zbl 1179.33020)], the notion of \(q\)-Dunkl operator is introduced and studied, which is a \(q\)-analogue of the well-known classical Dunkl operator. Several direct and inverse theorem of Jackson type are proved concerning the approximation of function from \(L^2_{q,\alpha} (\mathbb {R}_q)\), where the modulus of smoothness is constructed on the basis of \(q\)-Dunkl translation operator. As the approximation tool a class of functions with bounded spectrum is used, which is a class of functions for which their \(q\)-Dunkl transform are functions with compact support.
scientific article; zbMATH DE number 7379304