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Dunkl’s theory and best approximation by entire functions of exponential type in L 2-metric with power weight

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Abstract

In this paper, we study the sharp Jackson inequality for the best approximation of f ∈ L 2,κ(ℝd) by a subspace E 2 κ (σ) (SE 2 κ (σ)), which is a subspace of entire functions of exponential type (spherical exponential type) at most σ. Here L 2,κ(ℝd) 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 ⊂ ℝd and a function κ(ξ): R → ℝ+ invariant under the reflection group G(R) generated by R. In the case G(R) = ℤ d2 , we get some exact results. Moreover, the deviation of best approximation by the subspace E 2 κ (σ) (SE 2 κ (σ)) of some class of the smooth functions in the space L 2,κ(ℝd) is obtained.

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Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems, Ministry of Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P. R. China

    Yong Ping Liu & Chun Yuan Song

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  1. Yong Ping Liu
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  2. Chun Yuan Song
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Correspondence to Yong Ping Liu.

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Supported by National Natural Science Foundation of China (Grant No. 11071019), the research Fund for the Doctoral Program of Higher Education and Beijing Natural Science Foundation (Grant No. 1102011)

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Liu, Y.P., Song, C.Y. Dunkl’s theory and best approximation by entire functions of exponential type in L 2-metric with power weight. Acta. Math. Sin.-English Ser. 30, 1748–1762 (2014). https://doi.org/10.1007/s10114-014-2415-1

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  • DOI: https://doi.org/10.1007/s10114-014-2415-1

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