Journal of Approximation Theory
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 anyx∈Xfrom the setK≔C∩A−1(b),whereCis a closed convex subset ofX,Ais a bounded linearoperator fromXinto a finite-dimensional Hilbert spaceY, andb∈Y. 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.
Original Publication Citation
Deutsch, F., Li, W., & Ward, J. D. (1997). A dual approach to constrained interpolation from a convex subset of Hilbert space. Journal of Approximation Theory, 90(3), 385-414. doi:10.1006/jath.1996.3082
Deutsch, Frank; Li, Wu; and Ward, Joseph D., "A Dual Approach to Constrained Interpolation From a Convex Subset of Hilbert Space" (1997). Mathematics & Statistics Faculty Publications. 143.