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A Survey of Optimal Recovery

  • C. A. Micchelli
  • T. J. Rivlin
Part of the The IBM Research Symposia Series book series (IRSS)

Abstract

The problem of optimal recovery is that of approximating as effectively as possible a given map of any function known to belong to a certain class from limited, and possibly error-contaminated, information about it. In this selective survey we describe some general results and give many examples of optimal recovery.

Keywords

Hilbert Space Optimal Algorithm Linear Functional Feasibility Condition Optimal Recovery 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bakhvalov, N. S., On the optimality of linear methods for operator approximation in convex classes of functions, USSR Computational Mathematics and Mathematical Physics 11 (1971), 244–249.CrossRefGoogle Scholar
  2. 2.
    Bojanov, B.D., Optimal methods of interpolation in Comptes Rendus de l’Acadamie Bulgare des Sciences 27 (1974), 885–888.Google Scholar
  3. 3.
    Bojanov, B.D., Best methods of interpolation for certain classes of differentiable functions, Matematicheskie Zametki 17 (1975), 511–524.Google Scholar
  4. 4.
    Bojanov, B.D., Favard’s interpolation and best approximation of periodic functions, preprint.Google Scholar
  5. 5.
    deBoor, C., A remark concerning perfect splines, Bull. Amer. Math. Soc. 80 (1974), 724–727.CrossRefGoogle Scholar
  6. 6.
    Burchard, H., Interpolation and approximation by generalized convex functions, Ph.D., Dissertation, Purdue University, Lafayette, Indiana, 1968.Google Scholar
  7. 7.
    Caratheodory, C., Theory of Functions of a Complex Variable, Volume Two, 2nd English Edition, Chelsea, New York, 1960.Google Scholar
  8. 8.
    Danskin, John M., The theory of max-min, with applications. SIAM Journal, 14 (1966), 641–664.Google Scholar
  9. 9.
    Duren, Peter L., Theory of spaces, Academic Press, New York 1970.Google Scholar
  10. 10.
    Fisher, S.D. and J.W. Jerome, The existence, characterization and essential uniqueness of solutions of L extremal problems Trans. Amer. Math. Soc. 187 (1974), 391–404.Google Scholar
  11. 11.
    Gaffney, P.W. and M.J.D. Powell, Optimal Interpolation, C.S.S. 16 Computer Science and Systems Division, A.E.R.E., Harwell, Oxfordshire, England 1975.Google Scholar
  12. 12.
    Golomb, M., and H.F. Weinberger, Optimal approximation and error bounds, in On Numerical Approximation, R.E. Langer ed. The University of Wisconsin Press, Madison (1959), 117–190.Google Scholar
  13. 13.
    Golusin, G.M. Geometrische Funktionentheorie, VEB, Berlin 1957.Google Scholar
  14. 14.
    Holmes, R.B., A Course on Optimization and Best Approximation Lecture Notes Series 257, Springer-Verlag, Berlin 1972.Google Scholar
  15. 15.
    Karlin, S., Interpolation properties of generalized perfect splines and the solution of certain extremal problems I, Trans. Amer. Math. Soc. 206 (1975), 25–66.CrossRefGoogle Scholar
  16. 16.
    Karlin, S. and W.J. Studden, Tchebycheff Systems with Applications in Analysis and Statistics, Interscience Publishers, New York 1966.Google Scholar
  17. 17.
    Karlovitz, L.A., Remarks on variational characterization of Eigenvalues and n-widths problems, J. Math. Anal. Appl. 53 (1976), 99–110.CrossRefGoogle Scholar
  18. 18.
    Krein, M.G., The L-problem in abstract linear normed space in some questions in the theory of moments (N.I. Ahiezer, M.G. Krein, Eds.), Translations of Mathematical Monographs, Vol. 2 Amer. Math. Soc. Providence, R.I., 1962.Google Scholar
  19. 19.
    Krein, M.G., The ideas of P.L. Chebyshev and A.A. Markov in the theory of limiting values of integrals and their further developments, Amer. Math. Soc. Transl. 12 (1951), 1–122.Google Scholar
  20. 20.
    Marchuk, A.G., K. Yu Osipenko, Best approximation of functions specified with an error at a finite number of points, Matemati-cheskie Zametki 17 (1975), 359–368.Google Scholar
  21. 21.
    Meinguet, J., Optimal approximation and error bounds in semi-no rmed spaces, Numer. Math. 10 (1967) 370–388.Google Scholar
  22. 22.
    Melkman, A.A., n-widths and optimal interpolation of time-and band-limited functions, these proceedings.Google Scholar
  23. 23.
    Melkman, A.A. and C.A. Micchelli, On nonuniqueness of optimal subspaces for L n-width, IBM Research Report 6113 (1976).Google Scholar
  24. 24.
    Micchelli, C.A., Saturation classes and iterates of operators, Ph.D. Dissertation, Stanford University, 1969.Google Scholar
  25. 25.
    Micchelli, C.A., On an optimal method for the numerical differentiation of smooth functions, J. Approx. Theory 18(1976)189–204.Google Scholar
  26. 26.
    Micchelli, C.A., Best L1-approximation by weak Chebyshev systems and the uniqueness of interpolating perfect splines, to appear in J. Approx. Theory.Google Scholar
  27. 27.
    Micchelli, C.A., Optimal estimation of linear functionals, IBM Research Report 5729 (1975).Google Scholar
  28. 28.
    Micchelli, C.A., Optimal estimation of smooth functions from inaccurate data, in preparation.Google Scholar
  29. 29.
    Micchelli, C.A. and W. Miranker, High order search methods for finding roots, J. of Assoc. of Comp. Mach. 22 (1975), 51–60.CrossRefGoogle Scholar
  30. 30.
    Micchelli, C.A. and A. Pinkus, On n-widths in L, to appear Trans. Amer. Math. Soc.Google Scholar
  31. 31.
    Micchelli, C.A. and A. Pinkus, Moment theory for weak Chebyshev systems with applications to monosplines, quadrature formulae and best one-sided L1-approximation by spline functions with fixed knots, to appear in SIAM J. of Math. Anal.Google Scholar
  32. 32.
    Micchelli, C.A. and A. Pinkus, Total positivity and the exact n-width of certain sets in iX, to appear in Pacific Journal of Mathematics.Google Scholar
  33. 33.
    Micchelli, C.A. and A. Pinkus, On a best estimator for the class M using only function values, Math. Research Center, Univ. of Wisconsin, Report 1621 (1976).Google Scholar
  34. 34.
    Micchelli, C.A., and A. Pinkus, Some problems in the approximation of functions of two variables and the n-widths of integral operators, to appear as Math. Research Center Report, University of Wisconsin.Google Scholar
  35. 35.
    Micchelli, C.A., T.J. Rivlin, S. Winograd, Optimal recovery of smooth functions, Numer. Math. 260 (1976), 191–200.Google Scholar
  36. 36.
    Morozov, V.A. and A.L. Grebennikov, On optimal approximation of operators, Soviet Math Dokl. 16 (1975), 1084–1088.Google Scholar
  37. 37.
    Newman, D., Numerical differentiation of smooth data, preprint.Google Scholar
  38. 38.
    Osipenko, K. Yu, Optimal interpolation of analytic functions, Mathematicheskie Zametki 12 (1972), 465–476.Google Scholar
  39. 39.
    Osipenko, K. Yu, Best approximation of analytic functions from information about their values at a finite number of points, Matematischeski Zametki 19 (1976), 29–40.Google Scholar
  40. 40.
    Royden, H.L., Real Analysis, MacMillan Company, New York 1963.Google Scholar
  41. 41.
    Rudin, Walter, Functional Analysis, McGraw-Hill, New York 1973.Google Scholar
  42. 42.
    Sard, A., Optimal approximation, J. Funct. Anal. 1 (1967), 222–244.Google Scholar
  43. Sard, A., Addendum 2 (1968), 368–369.Google Scholar
  44. 43.
    Schultz, M.H., Complexity and differential equations, in Analytic Computational Complexity, J.F. Traub, Academic Press 1976.Google Scholar
  45. 44.
    Smolyak, S.A., On an optimal restoration of functions and functionals of them, Candidate Dissertation, Moscow State University 1965.Google Scholar
  46. 45.
    Schoenberg, I.J., The elementary cases of Landau’s problems of inequalities between derivatives, Amer. Math. Monthly 80 (1973), 121–158.CrossRefGoogle Scholar
  47. 46.
    Tihomirov, V.M., Best methods of approximation and interpolation of differentiable functions in the space C[-l,+l], Math. USSR Sbornik, 9 (1969), 275–289.CrossRefGoogle Scholar
  48. 47.
    Weinberger, H.F., On optimal numerical solution of partial differential equations, SIAM J. Numer. Anal. 9 (1972), 182–198.CrossRefGoogle Scholar
  49. 48.
    Wiener, N., Extrapolation, Interpolation, and Smoothing of Stationary Time Series, the Technology Press of MIT and J. Wiley New York, 1950.Google Scholar
  50. 49.
    Winograd, S., Some remarks on proof techniques in analytic complexity, in Analytic Computational Complexity, Ed., J.F. Traub, Academic Press, 1976.Google Scholar

Copyright information

© Springer Science+Business Media New York 1977

Authors and Affiliations

  • C. A. Micchelli
    • 1
  • T. J. Rivlin
    • 1
  1. 1.Research Staff Members of IBMIBMYorktown HeightsUSA

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