Chebyshev Polynomial Fitting for Satellite Precise Ephemerides

Chebyshev polynomials are widely employed for function approximation over defined intervals and are particularly effective for fitting satellite precise ephemerides, which provide critical data for determining orbital positions and velocities of satellites. The Chebyshev polynomial fitting process minimizes the difference between the target function and its polynomial approximation through mathematical optimization. This technique finds applications across numerous scientific and engineering domains, including weather pattern modeling and complex system behavior prediction. In practical implementation, the fitting typically involves these key steps: First, the ephemeris data is normalized to the interval [-1,1] to leverage Chebyshev polynomials' optimal properties. The algorithm then computes polynomial coefficients using least squares minimization, often implemented through numerical libraries like NumPy's polynomial.chebfit() function in Python or specialized orbit determination toolkits. The resulting compact representation significantly reduces storage requirements while maintaining high precision - typically achieving centimeter-level accuracy for GNSS satellites with just 10-15 coefficients per coordinate axis over 2-4 hour intervals. For code implementation, developers commonly utilize recurrence relations T_n(x) = 2xT_{n-1}(x) - T_{n-2}(x) for efficient polynomial evaluation, with careful attention to time segmentation and boundary condition handling to ensure continuity between adjacent ephemeris segments. This mathematical approach has established Chebyshev polynomial fitting as an essential tool in applied mathematics, particularly in space operations and satellite navigation systems where computational efficiency and accuracy are paramount.