No abstract

We show that a Banach space has modulus of convexity of power type p if and only if best approximants to points from straight lines are uniformly strongly unique of order p. Assuming that the space is smooth, we derive a characterisation of the best simultaneous approximant to two elements, and use the characterisation to prove that p-type modulus of convexity implies order p strong unicity of the simultaneous approximant.The study of strong unicity, initiated by Newman and Shapiro [9], resulted from a concern with the conditioning of the best approximation problem. In the present paper, the relation between strong unicity and modulus of convexity is explored. Throughout our discussion, we suppose that (X, ||-||) is a Banach space with modulus of convexity 8 > 0, and that V is a closed subspace of X. Bjornestal [2] showed that if x 0 G S(X) and if ||zo -2/|| ^ ll z o|| for every y G V, then, for every y G V with ||y|| ^ 3, that is, the vector 0 is a strongly unique best approximation to xo from V. We shall now show that Bjornestal's inequality characterises the modulus of convexity in a certain context, and that an analogue of this inequality holds in the non-uniformly convex space which underlies the theory of simultaneous approximation. We hope that these results will contribute to a deeper understanding of uniform convexity, as well as to the more "practical" goal of finding conditions under which tlie best approximation operator is continuous.We begin with precise definitions of the relevant concepts. Suppose that f £ X is fixed. An element z G V is said to be a best \\-\\-approximant to / from V if \\z -f\\ = inf{||w -f\\ : w G V}. Since X is uniformly convex, there is exactly one such z, and we denote it by /*. We say that /* is (M,j)-strongly unique of order p if, for some M -M(f) > 0, there exists a 7 = y(f,M) > 0 such that, for all v G V with | | » -/ ' | | < M

Let fJ denote the closed interval [0, 1] and let bA denote the set of all bounded, approximately continuous functions on n. Let Q denote the Banach space (sup noim) of quasi-continuous functions on fi. Let M denote the closed convex cone in Q comprised of non-decreasing functions. Let h p , 1 < p < oo, denote the best i p -simultaneaous approximation to the bounded measurable functions / and g by elements of M. It is shown that if / and g are elements of Q, then h p converges uniformly to a best L\ -simultaneous approximation of / and g. We also show that if / and g are in bA, then h p is continuous.

Suppose that f, g € L∞ [0,1 ] have discontinuities of the first kind only. Using the measure, max{ ∥f — h∥p, ∥g — h∥p}, of simultaneous Lp approximation, we show that the best simultaneous approximations f and g by nondecreasing functions converge uniformly as p → ∞. Part of the proof involves a discussion of discrete simultaneous approximation in a general context. Assuming only that f and g are approximately continuous, we show that their simultaneous best monotone Lp approximation is continuous.

Let <£ denote the Banach space (under the sup norm) of quasi-continuous functions on the unit interval [0, 1]. Let J! denote the closed convex cone comprised of monotone nondecreasing functions on [ 0 , 1 ] .