Abstract
In this thesis we consider polynomial approximations to functions, based on a criterion of F.Y.Edgeworth. The function to be approximated may be defined over a continuous interval or in a discrete set, and Edgeworth's criterion is that the aggregate (integral or sum) of the absolute error is to be minimized. This method of approximation is different from those associated with the names of Legendre (the method of least squares) and Chebyshev, and there are circumstances in which it is preferable to either of these. While ah exact solution of the problem is not always possible, yet there are several ways in which an approximate result may be arrived at. We consider both exact and approximate solutions. The first part of the thesis consists of an examination of the problem in cases where the data is specified over a continuous interval. Where appropriate, analogies are drawn to other methods of approximation, e.g. the Legendre and Chebyshev approaches mentioned above. The second part of the thesis deals with discrete data. The original method of Edgeworth is examined but abandoned as a practical method when the degree of the polynomial approximation desired exceeds unity; iterative methods are advocated as a better way of obtaining a solution in this case. In two chapters the relation between the continuous data problem and the discrete data problem is discussed. The thesis also contains an appendix consisting of relevant formulae and tables. Numerical examples are given throughout; the print-outs of two computer programmes are also included. A substantial part of this thesis consists of results which the author believes to be new.