Discrete orthogonal polynomials as a tool for detection of small anomalies of time series: a case study of GPS final orbits

Abstract: 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.