Lp approximation by quasi-convex and convex functions

Abstract

In this article a problem of approximation from nonconvex sets is considered. Let L p , 1 ⩽ p ⩽ ∞, be the Lebesgue space of extended real functions on a compact real interval. Given a subset P of quasi-convex functions and a function ƒ in L p , the problem is to find a best L p -approximation to ƒ from P ∩ L p . It is shown that if P is closed under pointwise convergence of sequences of functions, then a best approximation exists. Also investigated are properties of norm bounded subsets and convergent sequences of quasi-convex functions. Since convex and monotone functions are quasi-convex, the results are applicable to the problems of approximation from subsets of convex and monotone functions; in particular, the convex problem is analyzed in some detail.