论文标题
三角多项式对单数度量的近似和插值
Approximation and Interpolation of Singular Measures by Trigonometric Polynomials
论文作者
论文摘要
在许多应用程序中，有限总变化的复杂签名度量是一个强大的信号模型。限制在$ d $维的圆环中，如果已知的三角矩，则有限支持的措施允许精确恢复。在这里，我们考虑了一般度量的近似值，例如，通过相对于Wasserstein-1距离的固定程度的三角学多项式支持在曲线上支持。我们证明了它们的最佳近似值，并且（几乎）匹配的上限是可有效计算的近似值时，当该测量的三角学矩是有效的，可有效地计算上限。显示第二类正方形多项式总和可以插入量度支持上的特征函数，并收敛到外部零。
Complex signed measures of finite total variation are a powerful signal model in many applications. Restricting to the $d$-dimensional torus, finitely supported measures allow for exact recovery if the trigonometric moments up to some order are known. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the Wasserstein-1 distance. We prove sharp lower bounds for their best approximation and (almost) matching upper bounds for effectively computable approximations when the trigonometric moments of the measure are known. A second class of sum of squares polynomials is shown to interpolate the characteristic function on the support of the measure and to converge to zero outside.
