A Survey of Optimal Recovery

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.

[1]  J. Jerome,et al.  The existence, characterization and essential uniqueness of solutions of ^{∞} extremal problems , 1974 .

[2]  Charles A. Micchelli,et al.  High Order Search Methods for Finding Roots , 1975, JACM.

[3]  N. S. Bakhvalov,et al.  On the optimality of linear methods for operator approximation in convex classes of functions , 1971 .

[4]  T. J. Rivlin,et al.  The optimal recovery of smooth functions , 1976 .

[5]  Norbert Wiener,et al.  Extrapolation, Interpolation, and Smoothing of Stationary Time Series , 1964 .

[6]  Jean Meinguet,et al.  Optimal approximation and error bounds in seminormed spaces , 1967 .

[7]  Hermann Georg Burchard Interpolation and approximation by generalized convex functions , 1968 .

[8]  H. F. Weinberger On Optimal Numerical Solution of Partial Differential Equations , 1972 .

[9]  B. D. Boyanov Best methods of interpolation for certain classes of differentiable functions , 1975 .

[10]  K. Yu. Osipenko Optimal interpolation of analytic functions , 1972 .

[11]  Charles A. Micchelli,et al.  Moment Theory for Weak Chebyshev Systems with Applications to Monosplines, Quadrature Formulae and Best One-Sided $L^1 $-Approximation by Spline Functions with Fixed Knots , 1977 .

[12]  K. Yu. Osipenko Best approximation of analytic functions from information about their values at a finite number of points , 1976 .

[13]  Joseph F. Traub Analytic Computational Complexity , 1976 .

[14]  V M Tihomirov,et al.  BEST METHODS OF APPROXIMATION AND INTERPOLATION OF DIFFERENTIABLE FUNCTIONS IN THE SPACE $ C_{\lbrack-1,\,1\rbrack}$ , 1969 .

[15]  Charles A. Micchelli,et al.  Best L1 approximation by weak chebyshev systems and the uniqueness of interpolating perfect splines , 1977 .

[16]  L. Karlovitz,et al.  Remarks on variational characterizations of eigenvalues and n-width problems☆ , 1976 .

[17]  M. H. Schultz,et al.  Complexity and Differential Equations , 1976 .

[18]  K. Osipenko,et al.  Best approximation of functions specified with an error at a finite number of points , 1975 .

[19]  S. Winograd SOME REMARKS ON PROOF TECHNIQUES IN ANALYTIC COMPLEXITY , 1976 .

[20]  Samuel Karlin,et al.  Interpolation properties of generalized perfect splines and the solutions of certain extremal problems. I , 1975 .

[21]  Charles A. Micchelli On an optimal method for the numerical differentiation of smooth functions , 1976 .

[22]  I. J. Schoenberg The Elementary Cases of Landau's Problem of Inequalities Between Derivatives , 1973 .

[23]  Carl de Boor,et al.  A remark concerning perfect splines , 1974 .

[24]  J. Danskin The Theory of Max-Min, with Applications , 1966 .

[25]  Charles A. Micchelli,et al.  Some Problems in the Approximation of Functions of Two Variables and n-Widths of Integral Operators , 1978 .

[26]  Rudolph E. Langer,et al.  On Numerical Approximation , 1959 .

[27]  W. J. Studden,et al.  Tchebycheff Systems: With Applications in Analysis and Statistics. , 1967 .

[28]  Charles A. Micchelli,et al.  Total positivity and the exact $n$-width of certain sets in $L^1$. , 1977 .