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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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