A dual approach to constrained interpolation from a convex subset of hilbert space
Academic Article
Overview
Identity
Additional Document Info
Other
View All
Overview
abstract
Many interesting and important problems of best approximationare included in (or can be reduced to) one of the followingtype: in a Hilbert spaceX, find the best approximationPK(x) to anyxXfrom the setKCA-1(b),whereCis a closed convex subset ofX,Ais a bounded linearoperator fromXinto a finite-dimensional Hilbert spaceY, andbY. The main point of this paper is to show thatPK(x)isidenticaltoPC(x+A*y) - the best approximationto a certain perturbationx+A*yofx - from the convexsetCor from a certain convex extremal subsetCbofC. Thelatter best approximation is generally much easier to computethan the former. Prior to this, the result had been known onlyin the case of a convex cone or forspecialdata sets associatedwith a closed convex set. In fact, we give anintrinsic characterizationof those pairs of setsCandA-1(b) for which this canalways be done. Finally, in many cases, the best approximationPC(x+A*y) can be obtained numerically from existingalgorithms or from modifications to existing algorithms. Wegive such an algorithm and prove its convergence 1997 Academic Press.
