We show that polynomials do not belong to the reproducing kernel Hilbert space of infinitely differentiable translation-invariant kernels whose spectral measures have moments corresponding to a determinate moment problem. Our proof is based on relating this question to the problem of best linear estimation in continuous time one-parameter regression models with a stationary error process defined by the kernel. In particular, we show that the existence of a sequence of estimators with variances converging to $0$ implies that the regression function cannot be an element of the reproducing kernel Hilbert space. This question is then related to the determinacy of the Hamburger moment problem for the spectral measure corresponding to the kernel. In the literature it was observed that a non-vanishing constant function does not belong to the reproducing kernel Hilbert space associated with the Gaussian kernel (see Corollary 4.44 in Steinwart and Christmann, 2008). Our results provide a unifying view of this phenomenon and show that the mentioned result can be extended for arbitrary polynomials and a broad class of translation-invariant kernels.

翻译：我们的证据表明,多元分子并不属于由无限差异的翻译变异内核空间复制的内核,这些内核的光谱度量与确定时的问题相对应。我们的证据是,这一问题与连续时间最佳线性估计问题有关,一个参数回归模型具有由内核定义的固定错误过程。特别是,我们表明,存在一系列有差异的测算器,其差异趋同到0.美元,意味着回归功能不能成为再生产内核Hilbert空间的一个要素。然后,这一问题与与内核对应的光度测量的汉堡时点问题的确定性有关。在文献中,观察到,一个非蒸发性的恒定函数不属于与高斯内核相关的再生产内核的Hilbert空间(见Steinwart和Christmann的Colololorary 4.44,2008年)。我们的结果提供了这一现象的统一观点,并表明,所述结果可扩展为任意的多式多式和广型变异式。