A Dual Approach to Constrained Interpolationfrom a Convex Subset of Hilbert Space |
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Authors: | Frank Deutsch Wu Li Joseph D Ward |
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Institution: | aDepartment of Mathematics, The Pennsylvania State University, University Park, Pennsylvania, 16802;bDepartment of Mathematics, Old Dominion University, Norfolk, Virginia, 23529;cDepartment of Mathematics, Texas A &; M University, College Station, Texas, 77843 |
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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 setKC∩A−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 |
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