Constrained best approximation in Hilbert space .3. Applications to n-convex functions
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This paper continues the study of best approximation in a Hilbert space 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 to the n-convex functions. A subtle separation theorem is utilized to significantly extend the results in  and to obtain new results even for the "classical" cone of nonnegative functions. It was shown in  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 subcone CF. We will show how to determine this subcone CF, give the precise condition characterizing when CF= C, and apply and strengthen these general results in the practically important case when C is the cone of n-convex functions in L2(a, b).