论文标题
离散正交多项式作为检测小时异常序列的工具：GPS最终轨道的案例研究
Discrete orthogonal polynomials as a tool for detection of small anomalies of time series: a case study of GPS final orbits
论文作者
论文摘要
在本文中，我们表明，足够高度的经典离散单变量多项式（即，在具有单位重量的等距晶格上的Hahn多项式）具有极高的值，其值在端点附近具有极小的值（我们将此属性均为“最低限度的快速衰落”。 GP和GLONASS卫星的最终轨道中的异常值。提到的算法可在https://github.com/sptsarev/high-deg-polynomial-fitting上找到。 这些结果似乎是新的。在文献中找不到它们在离散正交多项式的著名渐近理论框架中的解释。
In this paper, we show that the classical discrete orthogonal univariate polynomials (namely, Hahn polynomials on an equidistant lattice with unit weights) of sufficiently high degrees have extremely small values near the endpoints (we call this property as "rapid decay near the endpoints of the discrete lattice". We demonstrate the importance of the proved results applying polynomial least squares approximation for the detection of anomalous values in IGS final orbits for GPS and GLONASS satellites. We propose a numerically stable method for the construction of discrete orthogonal polynomials of high degrees. It allows one to reliably construct Hahn-Chebyshev polynomials using standard accuracy (double precision, 8-byte) on thousands of points, for degrees up to several hundred. A Julia implementation of the mentioned algorithms is available at https://github.com/sptsarev/high-deg-polynomial-fitting. These results seem to be new; their explanation in the framework of the well-known asymptotic theory of discrete orthogonal polynomials could not be found in the literature.
