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 xk's, i.e., for points whose preimages on the circle by the application are almost equidistant. As it is well known [16, p. 99], , 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 qn of degree . On the other hand, if qn 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:
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.