In the present work we will address the problem of interpolating a
given continuous function *f* between *N*+1 distinct points
in
an interval [*a*,*b*]. We may choose
[*a*,*b*] := [-1,1] without loss of generality.

The classical solution is the interpolating polynomial *p* of degree ,
whose
determination is always a well-posed problem, but the use of which is
merely reasonable for
special choices of the *x*_{k}'s, i.e., for points whose preimages
on the circle by the
application
are almost equidistant. As it is well known
[16, p. 99], [12],
for equidistant points on the interval the polynomials *p* diverge or are
ill-conditioned as *N* increases.

But even with good points like Cebysev's, polynomial interpolation may not be
adequate. Markov's
inequality states that
for every polynomial
*q*_{n} of degree .
On the other hand, if *q*_{n} is a good approximation to *f*,
then
,
*e* small, and thus
;
therefore, no function
*f* with a derivative much larger than
at some point can be
well approximated by a polynomial of degree *n* (in the sense that
its derivative is
also approximated well). In other words, for a good approximation of *f*'as well as
*f* the degree of the interpolating polynomial should be at least
.
If
,
then a simultaneous ``good'' approximation of *f* and *f*' requires working
with interpolating polynomials *p* of such a large degree
that this may be numerically prohibitive.

The next infinitely differentiable choice for such functions as well as for arbitrary nodes seems to be rational interpolation [10,6]. The classical rational interpolant can be computed in a finite number of operations, but it has several drawbacks:

- --
- at least for small
*N*, it does not always exist and, if it does, it may exhibit poles in the interval of interpolation, which is not acceptable for the approximation of a continuous function (see Example 4 in §4); - --
- in general these disadvantages disappear as
*N*increases, but when*N*becomes larger than about 100 the computation is often affected by instability as a consequence of smearing (see the examples for equidistant points in [6]).

As a way of approximating functions with large gradients we suggest here to replace the interpolating polynomial of degree with the quotient of a polynomial of degree and a polynomial of prescribed degree that diminishes the maximal error as much as possible.

2000-05-30