Introduction

In the present work we will address the problem of interpolating a given continuous function f between N+1 distinct points $x_0,x_1,\ldots,x_N$ 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 $\le N$, 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 $\arccos$ 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 $\Vert q_n'\Vert _\infty\le n^2 \Vert q_n\Vert _\infty$ for every polynomial q_n of degree $\le n$. On the other hand, if q_n is a good approximation to f, then $\Vert q_n\Vert _\infty = \Vert f\Vert _\infty + e$, e small, and thus $\Vert q_n'\Vert _\infty\le n^2 (\Vert f\Vert _\infty + e)$; therefore, no function f with a derivative much larger than $n^2\Vert f\Vert _\infty$ 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 $\sqrt{\Vert f'\Vert _\infty/\Vert f\Vert _\infty}$. If $\Vert f'\Vert _\infty \gg\Vert f\Vert _\infty$, 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:

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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);

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in general these disadvantages disappear as N increases, but when Nbecomes 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 $\le N$ with the quotient of a polynomial of degree $\le N$ and a polynomial of prescribed degree that diminishes the maximal error as much as possible.