A Dual Approach to Constrained Interpolationfrom a Convex Subset of Hilbert Space
✍ Scribed by Frank Deutsch; Wu Li; Joseph D Ward
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 437 KB
- Volume
- 90
- Category
- Article
- ISSN
- 0021-9045
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✦ Synopsis
Many interesting and important problems of best approximation are included in (or can be reduced to) one of the following type: in a Hilbert space X, find the best approximation P K (x) to any x # X from the set K :=C & A &1 (b), where C is a closed convex subset of X, A is a bounded linear operator from X into a finitedimensional Hilbert space Y, and b # Y. The main point of this paper is to show that P K (x) is identical to P C (x+A* y) the best approximation to a certain perturbation x+A* y of x from the convex set C or from a certain convex extremal subset C b of C. The latter best approximation is generally much easier to compute than the former. Prior to this, the result had been known only in the case of a convex cone or for special data sets associated with a closed convex set. In fact, we give an intrinsic characterization of those pairs of sets C and A &1 (b) for which this can always be done. Finally, in many cases, the best approximation P C (x+A* y) can be obtained numerically from existing algorithms or from modifications to existing algorithms. We give such an algorithm and prove its convergence.