Abstract. Kövari and Pommerenke [19], and Elliott [8], have shown that the truncated Faber series gives a polynomial approximation which (for practical values of the degree of the polynomial) is very close to the best approximation. In this paper we discuss efficient Fast Fourier Transform (FFT) and recursive methods for the computation of Faber polynomials, and point out that the FFT method described by Geddes [13], for computing Chebyshev coefficients can be generalized to compute Faber coefficients.We also give a corrected bound for the norm of the Faber projection (that given in Elliott [8], being unfortunately slightly in error) and very briefly discuss a possible extension of the method to the case when the mapping function, which is required to compute the Faber series, is not known explicitly.

This article starts with a brief introduction to neural networks for those unfamiliar with the basic concepts, together with a very brief overview of mathematical approaches to the subject. This is followed by a more detailed look at three areas of research which are of particular interest to numerical analysts.The first area is approximation theory. IfKis a compact set in ℝn, for somen, then it is proved that a semilinear feedforward network with one hidden layer can uniformly approximate any continuous function inC(K) to any required accuracy. A discussion of known results and open questions on the degree of approximation is included. We also consider the relevance of radial basis functions to neural networks.The second area considered is that of learning algorithms. A detailed analysis of one popular algorithm (the delta rule) will be given, indicating why one implementation leads to a stable numerical process, whereas an initially attractive variant (essentially a form of steepest descent) does not. Similar considerations apply to the backpropagation algorithm. The effect of filtering and other preprocessing of the input data will also be discussed systematically.Finally some applications of neural networks to numerical computation are considered.

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[1] The heating of directly transmitted ions at low Mach number, quasi-perpendicular collisionless shocks is rapid, greater than adiabatic, and exhibits a distinct T ? > T k anisotropy. In this paper we present a theoretical study of the evolution of the ion velocity distribution across a stationary one-dimensional perpendicular model shock profile. A Lagrangian/Hamiltonian formulation of the ion equations of motion is introduced. We argue that the classical statistical physics solution of Liouville's equation in terms of the energy (Hamiltonian) is not applicable to the case of a laminar perpendicular shock. Assuming a Maxwellian incident ion velocity distribution, it is possible to obtain the analytical form for the distribution through the shock in terms of functions of upstream parameters that are independent of the incident temperature. Unlike the classical Hamiltonian solution, we show that contours of equal phase space probability do not correspond to contours of equal energy. It is this property of the velocity distribution that makes anisotropic heating possible. We recover the observed results that the distribution is stretched across the magnetic field direction as it passes through the shock and that it rotates as a whole around the field in the downstream region. We are able to show that in the low-temperature limit, the shape of the distribution remains Gaussian but that this is not the case for higher temperatures. For this Gaussian approximation, lower and upper bounds for the variance of the downstream velocity and therefore the heating are obtained. An efficient method for the numerical computation of the distribution through the shock is proposed and evaluated for typical shock parameter combinations. The downstream behaviour of the distribution is also elucidated.

Algorithms are presented for numerical type (m, n) rational approximation (m->_n-1) which can be effectively applied either in general simply connected regions of the complex plane or, as a special case, on intervals of the real line. In the former case particularly, the methods are much more efficient than those previously proposed; but they are also competitive even for intervals.The approximations are based on the observation that the Faber transform of a rational function is itself rational, and are of two types, a class of Pad6-1ike approximants and approximations of rational Carath6odory-Fej6r type based on the singular value decomposition of a Hankel matrix of Faber coefficients which generalise those recently introduced for the unit disc by Trefethen (Numer. Math., 37 (1981), pp. 297-320). These latter, particularly, give rise to approximations which are sufficiently near to best for practical purposes.