We study worst-case optimal approximation of positive linear functionals in reproducing kernel Hilbert spaces induced by increasingly flat Gaussian kernels. This provides a new perspective and some generalisations to the problem of interpolation with increasingly flat radial basis functions. When the evaluation points are fixed and unisolvent, we show that the worst-case optimal method converges to a polynomial method. In an additional one-dimensional extension, we allow also the points to be selected optimally and show that in this case convergence is to the unique Gaussian quadrature–type method that achieves the maximal polynomial degree of exactness. The proofs are based on an explicit characterisation of the reproducing kernel Hilbert space of the Gaussian kernel in terms of exponentially damped polynomials.

中文翻译：

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最平坦的高斯核的最坏情况最佳逼近

我们研究了正线性函数在最坏情况下的最优逼近，这些正线性函数在由越来越平坦的高斯核引起的核Hilbert空间中。这为径向基函数越来越平坦的插值问题提供了新的视角和一些概括。当评估点固定且为单一溶剂时，我们表明最坏情况的最优方法收敛于多项式方法。在附加的一维扩展中，我们还允许最佳选择点，并表明在这种情况下，收敛是达到最大多项式精确度的唯一高斯正交类型方法。证明基于指数阻尼多项式对高斯核的再生核Hilbert空间的显式表征。