My original introduction to this subject was through conservations, and ultimate­ ly joint work with C. A. Micchelli. I am grateful to him and to Profs. C. de Boor, E. W. Cheney, S. D. Fisher and A. A. Melkman who read various portions of the manuscript and whose suggestions were most helpful. Errors in accuracy and omissions are totally my responsibility. I would like to express my appreciation to the SERC of Great Britain and to the Department of Mathematics of the University of Lancaster for the year spent there during which large portions of the manuscript were written, and also to the European Research Office of the U.S. Army for its financial support of my research endeavors. Thanks are also due to Marion Marks who typed portions of the manuscript. Haifa, 1984 Allan Pinkus Table of Contents 1 Chapter I. Introduction . . . . . . . . Chapter II. Basic Properties of n-Widths . 9 1. Properties of d . . . . . . . . . . 9 n 15 2. Existence of Optimal Subspaces for d . n n 17 3. Properties of d . . . . . . 20 4. Properties of b . . . . . . n 5. Inequalities Between n-Widths 22 n 6. Duality Between d and d . . 27 n 7. n-Widths of Mappings of the Unit Ball 29 8. Some Relationships Between dn(T), dn(T) and bn(T) . 32 37 Notes and References . . . . . . . . . . . . . .

Inhalt

I. Introduction.- II. Basic Properties of n-Widths.- 1. Properties of dn.- 2. Existence of Optimal Subspaces for dn.- 3. Properties of dn.- 4. Properties of ?n.- 5. Inequalities Between n-Widths.- 6. Duality Between dn and dn.- 7. n-Widths of Mappings of the Unit Ball.- 8. Some Relationships Between dn(T), dn(T) and ?n(T).- Notes and References.- III. Tchebycheff Systems and Total Positivity.- 1. Tchebycheff Systems.- 2. Matrices.- 3. Kernels.- 4. More on Kernels.- IV. n-Widths in Hilbert Spaces.- 1. Introduction.- 2. n-Widths of Compact Linear Operators.- 3. n-Widths, with Constraints.- 3.1 Restricted Approximating Subspaces.- 3.2 Restricting the Unit Ball and Optimal Recovery.- 3.3 n-Widths Under a Pair of Constraints.- 3.4 A Theorem of Ismagilov.- 4. n-Widths of Compact Periodic Convolution Operators.- 4.1 n-Widths as Fourier Coefficients.- 4.2 A Return to Ismagilov's Theorem.- 4.3 Bounded mth Modulus of Continuity.- 5. n-Widths of Totally Positive Operators in L2.- 5.1 The Main Theorem.- 5.2 Restricted Approximating Subspaces.- 6. Certain Classes of Periodic Functions.- 6.1 n-Widths of Cyclic Variation Diminishing Operators.- 6.2 n-Widths for Kernels Satisfying Property B.- Notes and References.- V. Exact n-Widths of Integral Operators.- 1. Introduction.- 2. Exact n-Widths of K? in Lq and Kp in L1.- 3. Exact n-Widths of K?r in Lq and Kpr in L1.- 4. Exact n-Widths for Periodic Functions.- 5. n-Widths of Rank n + 1 Kernels.- Notes and References.- VI. Matrices and n-Widths.- 1. Introduction and General Remarks.- 2. n-Widths of Diagonal Matrices.- 2.1 The Exact Solution for q ? p and p = 1, q = 2.- 2.2 Various Estimates for p = 1, q = ?.- 3. n-Widths of Strictly Totally Positive Matrices.- Notes and References.- VII. Asymptotic Estimates for n-Widths of Sobolev Spaces.- 1. Introduction.- 2. Optimal Lower Bounds.- 3. Optimal Upper Bounds.- 4. Another Look at ?n(B1(r); L?).- Notes and References.- VIII. n-Widths of Analytic Functions.- 1. Introduction.- 2. n-Widths of Analytic Functions with Bounded mth Derivative.- 3. n-Widths of Analytic Functions in H2.- 4. n-Widths of Analytic Functions in H?.- 5. n-Widths of a Class of Entire Functions.- Notes and References.- Glossary of Selected Symbols.- Author Index.