This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Andersson, J-E., Optimal quadrature of Hp functions, Math. Z. 172, 1980, 55–62.
Babadjanov, S. B. and V. M. Tichomirov, On the width of a functional class in the space Lp (p ≥ 1), Izv. Akad. Nauk UZSR, Ser. Fiz-Mat. Nauk 12, 1967, 24–30.
Berger, Melvin, and Marion Berger, "Perspectives in Nonlinearity", W. A. Benjamin, Inc., N. Y., 1968.
Bojanov, B. D., Best quadrature formula for a certain class of analytic functions, Zastos. Mat., XIV, 1974, 441–447.
Bojanov, B. D., Uniqueness of monosplines of least deviation, "Numerische Integration", Ed. G. Hämmerlin, International Series of Numerical Mathematics, Vol. 45, Birkhäuser, Basel, 1979, 67–97.
Braess, D., and N. Dyn, On the uniqueness of generalized monosplines with least Lp norm, to appear.
Cheney, E. W., "Introduction to Approximation Theory," McGraw-Hill Book Co., N. Y., 1966.
Dahmen, W., Micchelli, C. A. and P. W. Smith, Asymptotically optimal sampling schemes for periodic functions, IBM Research Report, RC 10733, 1984.
Davis, C., Kahan, W. M., and H. F. Weinberger, Norm-preserving dilations and their applications to optimal error bounds, SIAM J. Numer. Anal. 19, 1982, 445–469.
Duchon, J., Splines minimizing rotation-invariant semi-norms in Sobolev spaces, "Constructive Theory of Functions of Several Variables", Eds. W. Schempp and K. Zeller, Springer, Berlin-Heidelberg, 1976, 85–100.
Duren, P., "Theory of Hp Spaces", Academic Press, N. Y., 1970.
Fisher, S. D., and C. A. Micchelli, The n-width of sets of analytic functions, Duke Math. J. 47, 1980, 789–801.
Fisher, S. D., and C. A. Micchelli, Optimal sampling of holomorphic functions, Amer. J. Math. 106, 1984, 593–609.
Gaffney, P. W., and M.J.D. Powell, Optimal interpolation, "Numerical Analysis" (Proc. 6th Biennial Dundee Conf., Univ. Dundee, 1975). Lecture Notes in Math., Vol. 506, Springer, Berlin, 1976, 90–99.
Gal, S., and C. A. Micchelli, Optimal sequential and non-sequential procedures for evaluating a functional, Applicable Analysis 10, 1980, 105–120.
Gantmacher, F. R., and M. G. Krein, "Oscillationsmatrizen, Oscillationskerne und kleine Schwingungen mechanischer Systeme", Akademie-Verlag, Berlin, 1960.
Golomb, M. and Weinbeiger, H. F., Optimal approximation and error bounds, "On Numerical Approximation", Ed. R. Langer, U. of Wisconsin Press, Madison, Wisconsin, 1959, 117–190.
Höllig, K., A generalization of Jackson’s inequality, J. Approx. Theory 31, 1981, 154–157.
Krylov, V. I., "Approximate Calculation of Integrals", Macmillan, N. Y. 1962.
Kuo, H.-H., "Gaussian Measures in Banach Spaces", Lecture Notes in Mathematics, Vol. 463, Springer, Berlin, 1975.
Larkin, F. M., Gaussian measure in Hilbert space and application to numerical analysis, Rocky Mt. J. Math. 2, 1972, 379–421.
Laüter, H., A minmax linear estimator for linear parameters under restrictions in form of inequalities, Math. Operationsforsch. Statist. 6, 1975, 689–695.
Levin, M., and J. Girshovich, "Optimal Quadrature Formulas", B. G. Teubner, Leipzig, 1979.
Loeb, H. L., A note on optimal integration in H ∞, C. R. Akad. Bulgare Sci. 27, 1974, 615–618.
Logan, B. F., and L. A. Shepp, Optimal reconstruction of a function from its projections, Duke Math. J. 42, 1975, 645–659.
Makovoz, Yu., I., On a method for estimation from below of diameters of sets in Banach spaces, Math. U.S.S.R. Sb. 16, 1972, 139–146.
Meinguet, J., An intrinsic approach to multivariate spline interpolation at arbitrary points, "Polynomial and Spline Approximation", Ed., B. N. Sahney, D. Reidel, Dordrecht, 1979, 163–190.
Melkman, A. A., n-widths and optimal interpolation of time-and band-limited functions, "Optimal Estimation in Approximation Theory", Eds., C. A. Micchelli and T. J. Rivlin, Plenum Press, N. Y., 1977, 55–68.
Melkman, A. A. and C. A. Micchelli, Spline spaces are optimal for L2 n-widths, Illinois J. Math. 22, 1978, 541–564.
Melkman, A. A. and C. A. Micchelli, Optimal estimation of linear operators in Hilbert spaces from inaccurate data, SIAM J. Numer. Anal. 16, 1979, 87–105.
Melkman, A. A., n-widths and optimal interpolation of time-and band-limited functions II, SIAM J. Math. Anal., to appear.
Micchelli, C. A., Best quadrature formulas at equally spaced nodes, J. Math. Anal. Appl. 47, 1974, 232–249.
Micchelli, C. A., Optimal estimation of smooth functions from inaccurate data, J. Inst. Maths. Applics. 23, 1979, 473–495.
Micchelli, C. A., On Weinberger’s duality formula for the approximation of bounded operators on Hilbert spaces, IBM Research Report, RC #7722, 1979.
Micchelli, C. A., Orthogonal projections are optimal algorithms, J. Approx. Theory 40, 1984, 101–110.
Micchelli, C. A. and A. Pinkus, Some problems on the approximation of functions of two variables and n-widths of integral operators, J. Approx. Theory 24, 1978, 51–77.
Micchelli, C. A., T. J. Rivlin and S. Winograd, The optimal recovery of smooth functions, Numer. Math. 26, 1976, 191–200.
Micchelli, C. A. and T. J. Rivlin, A survey of optimal recovery, "Optimal Estimation in Approximation Theory" (Eds. C. A. Micchelli and T. J. Rivlin), Plenum, N. Y., 1977, 1–54.
Micchelli, C. A. and T. J. Rivlin, Optimal recovery of best approximations, Resultate der Math. 3, 1980, 25–32.
Micchelli, C. A., and G. Wahba, Design problems for optimal surface interpolation, "Approximation Theory and Applications", Ed. Z. Ziegler, Academic Press, N. Y., 1981, 329–348.
Mirsky, L., Symmetric gauge functions and unitarily invariant norms, Quart. J. Math. Oxford Ser. (2) 11, 1960, 50–59.
Newman, D. J., Rational approximation to |x|, Michigan Math. J. 11, 1964, 11–14.
Newman, D. J., Quadrature in Hp, Lectures III, IV, "Approximation with Rational Functions", CBMS Regional Conference Series in Math., No. 41, Amer. Math. Soc., Providence, Rhode Island, 1979.
Osipenko, K. Yu, Optimal interpolation of analytic functions, Math. Notes Acad. Sci. U.S.S.R. 12, 1972, 712–719.
Osipenko, K. Yu., Best methods of approximating analytic functions given with an error, Math. U.S.S.R. Sbornik 46, 1983, 353–374.
Pinkus, A., n-widths of Sobolev spaces in Lp, to appear in Constructive Approximation.
Rivlin, T. J., A survey of recent results on optimal recovery, "Polynomial and Spline Interpolation", Ed. B. N. Sahney, D. Reidel, Dordrecht 1979, 225–245.
Rivlin, T. J., The optimal recovery of functions, Contemporary Math., Vol. 9, 1982, 121–151.
Rivlin, T. J., St. Ruscheweyh, D. Shaffer and K. J. Wirths, Optimal recovery of the derivative of bounded analytic functions, IMA J. of Numer. Anal. 3, 1983, 327–332.
Rivlin, T. J., and D. B. Shaffer, Optimal estimation of the derivative of bounded analytic functions, IBM Research Report, RC #9843, 1983.
Scharlach, R., Optimal recovery by linear functionals, J. Approx. Theory, to appear.
Speckman, P., Minmax estimates of linear functionals in a Hilbert space, preprint.
Traub, J. F. and H. Wozniakowski, "A General Theory of Optimal Algorithms", Academic Press, N. Y., 1980.
Weinberger, H. F., On optimal numerical solution of partial differential equations, SIAM J. Numer. Anal. 9, 1972, 182–198.
Zensykbaev, A. A., Optimal recovery methods for the integral on classes of differentiable functions, Analysis Mathematica 7, 1981, 303–318.
Zhensykbaev, A. A., Extremal properties of monosplines and best quadrature formulas, Mathematical Notes 31, 1982, 145–154.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1985 Springer-Verlag
About this paper
Cite this paper
Micchelli, C.A., Rivlin, T.J. (1985). Lectures on optimal recovery. In: Turner, P.R. (eds) Numerical Analysis Lancaster 1984. Lecture Notes in Mathematics, vol 1129. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075157
Download citation
DOI: https://doi.org/10.1007/BFb0075157
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-15234-7
Online ISBN: 978-3-540-39412-9
eBook Packages: Springer Book Archive