BIT 17 (1977), 262-269 UNIFORM APPROXIMATION BY GENERALISED POLYNOMIALS M. BRANNIGAN Abstract. We here describe methods both numerical and analytic to solve the problem of finding the best uniform approximations to a continuous function by a finite dimensional linear space of functions which does not necessarily satisfy the Haar condition. We show how a knowledge of H-sets is essential and how the theory is simplified by the use of this concept. 1. Introduction. We consider the problem of approximating a continuous function fe C(B) where B is some compact metric space, by some linear subspace VcC(B) of :limensiofl say n, spanned by {gl, g2,...,g,,}. Using the uniform norm we are guaranteed existence [see theorem 1,14], however, in most practical case s uniqueness does not follow as V will in general not satisfy the Haar condition. The theory of approximation by such generalised polynomials, elements of F, is most advantageously studied using the concept of an H-set, which replaces in a natural way the "alternants" of Chebyshev systems. We here give a definition which is similar to that for the Haar condition and it is readily seen [1] to be equivalent to that of Collatz [15] and the "extremal signatures" of Rivlin and Shapiro [16]. DEFINITION. The set of points x 1, x 2 .... Xk e B form an H-set with respect to V if and only if there exist 2 i > 0 and e~= +_ 1, i= 1, 2,..., k satisfying We denote such an H-set by the quadruple [ x i, ,;t i, ei, k]. It is evident that k__< n + 1 and in the case where all H-sets are "maximal', that is k = n + 1, then we have a Chebyshev system. We will refer to an H-set being "minimal" if no proper subset of the {xi} form an H-set. The theoretical considerations are fully detailed in [1, 2, 5, 6, 7, 9, 10, 11, 16, 18]. In section 2 we consider the problem of finding the set of best approximations when uniqueness does not exist and in section 3 we formulate a numerical method based directly on the simultaneous exchange algorithm of Remez [15]. Received October 23, 1976. Revised April 23, 1977.