aequationes mathematicae

, Volume 31, Issue 1, pp 18–25 | Cite as

Linear problems (with extended range) have linear optimal algorithms

  • Edward W. Packel
Research Papers

Abstract

LetF1 andF2 be normed linear spaces andS:F0F2 a linear operator on a balanced subsetF0 ofF1. IfN denotes a finite dimensional linear information operator onF0, it is known that there need not be alinear algorithmφ:N(F4) →F2 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 imbeddingF2 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.

AMS (1980) subject classification

Primary 41A45, 41A65, 65J10 Secondary 46B99, 68C05 

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References

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    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.Google Scholar
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    Packel, E. W.,Functional analysis. A Short Course. Robert E. Krieger Publishing Co., Inc., Huntington, NY, 1980.Google Scholar
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    Smolyak, S. A.,On optimal restoration of functions and functionals of them (Russian). Candidate Disseration, Moscow State Univ., 1965.Google Scholar
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    Traub, J. F. andWoźniakowski, H.,A general theory of optimal algorithms. Academic Press, 1980.Google Scholar

Copyright information

© Birkhäuser Verlag 1986

Authors and Affiliations

  • Edward W. Packel
    • 1
  1. 1.Department of Computer ScienceColumbia UniversityNew YorkUSA

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