# Generalized order star theory

## Abstract

In this paper we generalize the theory of order stars of Wanner, Hairer and Nørsett [14]. We show that there is a geometric relation between the location of the zeros and the poles of a rational approximation to the exponential and the distribution of its interpolation points.

By applying this theory we find that the A-acceptability and the general form of the denominator impose bounds on the number and location of the interpolation points. These bounds are used to characterize the A-acceptability properties of various families of rational approximations to exp(x), in particular to verify a conjecture of Ehle [2] on the order-constrained Chebishev approximations.

Much of the paper is based upon a joint work of the author with M.J.D. Powell [8].

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