Linear problems (with extended range) have linear optimal algorithms
- Edward W. Packel
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LetF 1 andF 2 be normed linear spaces andS:F 0 →F 2 a linear operator on a balanced subsetF 0 ofF 1. IfN denotes a finite dimensional linear information operator onF 0, it is known that there need not be alinear algorithmφ:N(F 4) →F 2 which is optimal in the sense that ‖φ(N(f)) −S(f‖ is minimized. We show that the linear problem defined byS andN can be regarded as having a linear optimal algorithm if we allow the range ofφ to be extended in a natural way. The result depends upon imbeddingF 2 isometrically in the space of continuous functions on a compact Hausdorff spaceX. This is done by making use of a consequence of the classical Banach-Alaoglu theorem.
- Osipenko, K. Yu.,Best approximation of analytic functions from information about their values at a finite number of points (Russian). Math. Notes19 (1976), 17–23.
- Packel, E. W.,Functional analysis. A Short Course. Robert E. Krieger Publishing Co., Inc., Huntington, NY, 1980.
- Smolyak, S. A.,On optimal restoration of functions and functionals of them (Russian). Candidate Disseration, Moscow State Univ., 1965.
- Traub, J. F. andWoźniakowski, H.,A general theory of optimal algorithms. Academic Press, 1980.
- Linear problems (with extended range) have linear optimal algorithms
Volume 31, Issue 1 , pp 18-25
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- Edward W. Packel (1)
- Author Affiliations
- 1. Department of Computer Science, Columbia University, 10027, New York, N Y, USA