Constructive approximation / Ronald A. DeVore, George G. Lorentz

Includes bibliographical references (pages [434]-446) and index

Contents Ch. 1 Theorems of Weierstrass 1 Basic Notions 2 Approximation by Integral Operators 3 The Theorem of Korovkin 4 Theorems of Stone-Weierstrass Ch. 2 Spaces of Functions 1 Introduction. The Spaces C and L[subscript p] 2 Rearrangement-Invariant Function Spaces 3 Hardy's Inequalities and the ([theta],q)-quasi-norms 4 Linear Operators. Interpolation of Operators 5 Spaces of Differentiable Functions: Sobolev Spaces 6 Moduli of Continuity 7 Moduli of Smoothness 8 Marchaud Inequalities 9 Lipschitz Spaces 10 Besov Spaces Ch. 3 Best Approximation 1 Introduction: Existence of Best Approximation 2 Kolmogorov's Theorem 3 Haar Systems 4 Uniqueness of Best Approximation in C(A) 5 Chebyshev's Theorem 6 Chebyshev Polynomials 7 Strong Unicity 8 Remez Algorithms 9 Krein's Theorem 10 Best Approximation in L[subscript p], [actual symbol not reproducible] 11 Polya and Descartes Systems 12 Weak Haar Systems Ch. 4 Properties of Polynomials 1 Inequalities of Bernstein, Szego and Markov 2 Polynomials on the Complex Plane and in Banach Spaces 3 Bernstein Inequalities in L[subscript p], 0 < p < 1 4 Polynomials with Positive Coefficients in x, 1 - x 5 Lagrange Interpolation 6 Hermite Interpolation 7 Divided Differences 8 Quadrature Formulas 9 Birkhoff Interpolation 10 Regularity of Birkhoff Matrices Ch. 5 Splines 1 Definitions and Simple Properties 2 B-Splines 3 B-Spline Series 4 Quasi-Interpolant Operators 5 Euler and Bernoulli Splines 6 Definition of Splines by Their Extremal Properties 7 The Kolmogorov-Landau Inequalities 8 Zero Count for Splines 9 Spline Interpolation 10 Sign Variation of Splines 11 Total Positivity of the B-Spline Collocation Matrix Ch. 6 K-Functionals and Interpolation Spaces 1 K-Functionals 2 K-Functionals and Moduli of Smoothness 3 Comparisons of Moduli of Smoothness 4 Two Theorems of Whitney 5 Averaged Moduli of Smoothness 6 Moduli of Smoothness with Weights 7 The [theta],q-Interpolation Spaces Ch. 7 Central Theorems of Approximation 2 Trigonometric Approximation 3 Inverse Theorems of Trigonometric Approximation 4 Favard's Theorems 5 Improvement of Estimates 6 Approximation by Algebraic Polynomials 7 Spline Approximation 8 Approximation of Analytic Functions 9 Approximation Spaces Ch. 8 Influence of Endpoints in Polynomial Approximation 2 Local Inequalities for Polynomials 3 Properties of the Jackson Operators P[subscript n,m](f) 4 Simultaneous Approximation of Functions and Their Derivatives 5 Brudnyi's Theorem 6 Inverse Theorems 7 Approximation Spaces for Algebraic Polynomials Ch. 9 Approximation by Operators 2 Computation of Some Norms 3 Examples of Linear Polynomial Operators 4 Positive Operators 5 Projections onto Spaces Spanned by Exponentials 6 Lower Bounds 7 Projections in Arbitrary Banach Spaces 8 Families of Commuting Operators Ch. 10 Bernstein Polynomials 1 Definitions and Inequalities 2 Derivatives of Bernstein Polynomials 3 Approximation and Shape Preserving Properties 4 Bernstein Polynomials of Convex Functions 5 Saturation and Inverse Theorems 6 Saturation Theorems for Kantorovich Polynomials 7 Characterization of Approximation Spaces 8 Further Properties and Variants of Bernstein Polynomials Weak Haar Spaces Ch. 11 Approximation of Classes of Functions, Muntz Theorems 1 Approximation by Fourier Sums 2 Saturation Classes 3 Saturation of the Fejer Operators 4 Theorems of Korneichuk 5 Muntz' Theorem. Approximation of Monomials 6 Case When [actual symbol not reproducible]. Selection of Best Powers Ch. 12 Spline Approximation 2 Splines with Equally Spaced Knots 3 Approximation by Dyadic Splines 4 Splines with Free Knots 5 Smoothness in L[subscript p] for 0 < p < 1 6 Dyadic Splines in L[subscript p], 0 < p < 1 7 Comparison of the Spaces [actual symbol not reproducible] 8 Free Knot Spline Approximation in L[subscript p], 0 < p < [actual symbol not reproducible] Ch. 13 Spline Interpolation and Projections onto Spline Spaces 1 Introduction. Lagrange Interpolation by Splines 2 Selection of Interpolation Points 3 Cubic Spline Interpolation 4 Orthogonal Projection onto Splines 5 Interpolation on R 6 Cardinal Spline Interpolation 7 Approximation from Shift Invariant Spaces 8 Shape Preserving Interpolation 9 Shape Preserving Quadratic Spline Interpolation Bibliography Index