CONSTRAINED BEST APPROXIMATION IN HILBERT-SPACE .3. APPLICATIONS TO N-CONVEX FUNCTIONS

This paper continues the study of best approximation in a Hilbert spac e X from a subset K which is the intersection of a closed convex cone C and a closed linear variety, with special emphasis on applications t o the n-convex functions. A subtle separation theorem is utilized to s ignificantly extend the results in [4] and to obtain new results even for the ''classical'' cone of nonnegative functions. It was shown in [ 4] that finding best approximations in K to any f in X can be reduced to the (generally much simpler) problem of finding best approximations to a certain perturbation of f from either the cone C or a certain su bcone C-F. We will show how to determine this subcone C-F, give the pr ecise condition characterizing when C-F = C, and apply and strengthen these general results in the practically important case when C is the cone of n-convex functions in L(2)(a, b).