THE EXPERIENCE OF NUMERICAL INTEGRATION AND APPROXIMATION WITH APPLYING CHEBYSHEV POLYNOMIALS FOR CONSTRUCTING EPHEMERIDES OF THE SOLAR SYSTEM NATURAL AND ARTIFICIAL BODIES Anna A. Trubitsina Institute of Theoretical Astronomy Russian Academy of Sciences Naberezhnaya Kutuzova 10, St.Petersburg, 191187 Russia E-mail 1104@ita.spb.su The successful experience of applying the Chebyshev polynomials as a power "mathematical tool" for numerical integration and approximation techniques in celestial mechanics is presented. The results of series of testing the single-step integrator INCH with uniform approximation of solution by Chebyshev expansions have demonstrated its high efficiency for numerical solving various astronomical problems. This procedure was used for computation of the fundamental ephemerides AE of the Sun, Moon and major planets of the Solar system, various Earth's artificial satellite orbits as well as orbits of natural satellites of planets. The detailed analysis of approximation function behavior inside of integration step has allow to elaborate a special modification of INCH procedure for high accuracy and rapid integration of piece-wise continuous functions, modeling the Earth's shadow effect for artificial satellite orbits. The developed algorithm of the Chebyshev polynomials representation of ephemeris data makes it possible to compress essentially the initial information by representing it in a compact form with same accuracy of solution. The original software has been elaborated for calculating ephemeris file simultaneously with the process of numerical integration of orbits by the procedure INCH. This technique has been applied for the construction of ephemerides of the natural and artificial celestial bodies as well as for the compact polynomial representation of different geodynamical parameters e.g. new nutation theories. The possible new applications of elaborated algorithms are discussed in conclusion.