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Dunkl’s theory and best approximation by entire functions of exponential type in \(L_{2}\)-metric with power weight. (English) Zbl 1304.41020
Summary: In this paper, we study the sharp Jackson inequality for the best approximation of \(f \in L_{2,{\kappa}}(\mathbb{R}^{d})\) by a subspace \(E_{\kappa}^{2}(\sigma)\, (SE_{\kappa}^{2}(\sigma))\), which is a subspace of entire functions of exponential type (spherical exponential type) at most \(\sigma\). Here \(L_{2,\kappa}(\mathbb{R}^{d})\) denotes the space of all \(d\)-variate functions \(f\) endowed with the \(L_{2}\)-norm with the weight \(v_\kappa (x) = \prod\nolimits_{\xi \in R_ + } {|(\xi ,x)|^{2\kappa (\xi)} } \), which is defined by a positive subsystem \(R_{+}\) of a finite root system \(R \subset \mathbb{R}^d\) and a function \(\kappa(\xi): R \to \mathbb{R}_{+}\) invariant under the reflection group \(G(R)\) generated by \(R\). In the case \(G(R) = \mathbb{Z}_{2}^{d}\), we get some exact results. Moreover, the deviation of best approximation by the subspace \(E_{\kappa}^{2}(\sigma)\, (SE_{\kappa}^{2}(\sigma))\) of some class of the smooth functions in the space \(L_{2,{\kappa}}(\mathbb{R}^{d})\) is obtained.
MSC:
41A50 Best approximation, Chebyshev systems
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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