Astronomers are supplied with ephemerides coming either from analytical theories or from the numerical integration of the equations of motion. In both cases the astronomical ephemerides are usually provided in tabular form. For points not listed in the tables the user is forced to interpolate. There is another method for providing ephemerides: for any interval of the data base a set of coefficients enables the construction of a polynomial representation from which the user can recover the data base (within a certain precision).

When using polynomial approximations it arises the problem of establishing a bound for the error. For instance, the error coming from a least squares fit varies in a nonuniform way and, hence, it is not possible to make a previous estimation of the error. This inconvenience is avoided when using uniform approximation: the extreme values of the error are equal in absolute value. We describe the method in Sect. 2 (click here).

Fitting astronomical ephemerides with polynomial approximations is not a new affair; for instance, since 1980 in "Connaisance des Temps'' the Bureau des Longitudes computes series expansions in Chebyshev polynomials representing their analytical theories. Alternative procedures to compress planetary and satellite ephemerides keeping a good precision have also been proposed (Chapront & Vu 1984; Chapront 1995).

When an analytical function is approximated by means of a polynomial, fast evaluation is ensured for any point of the interval of validity. The approximation of discrete functions is a different matter. At first one should only claim to recover the data base from the polynomial approximation but in many cases it can be also used for interpolation between points of the data base.

In this work we approximate discrete functions finding the *best*
approximation in the Chebyshev sense. We apply this uniform
approximation to the Jet Propulsion Laboratory DE200/LE200 planetary and
lunar ephemerides (Newhall et al. 1983). It must be mentioned that
the Nautical Almanac Office of the U.S. Naval Observatory issued the
"Almanac for Computers'' from 1977 (Kaplan et al. 1977)
to 1991, where standard Chebyshev approximation was used to compute polynomials
covering specified intervals of time (Kaplan, private communication).
These polynomial approximations are used to produce ephemerides with lower
precision than DE200/LE200. Beyond 1991 that publication was replaced by
integrated packages of software and astronomical data.

In our approach using the best approximation we obtained medium or low degree polynomial valid for long time intervals except for the moon. In Sect. 3 (click here) comparisons with standard Chebyshev approximation and least squares fit are carried out. Best results were always on the side of the best approximation.

In Sect. 4 (click here) the best approximation is used for construction of the ANDI (Lara & López 1997), an electronical nautical almanac where fast evaluation is required. In order to overcome the inconvenience that the estimator of the error is only valid for the set of points used to construct the approximation, we look back to the best approximation to continuous functions, where the estimate for the error fits for all the interval of validity. Working with dense data bases, the smoothness of the resulting discrete error functions guarantees the reliable use of the approximating polynomials like interpolating ones.