Preface
1 The approximation problem and existence of best approximations
1.1 Examples of approximation problems
1.2 Approximation in a metric space
1.3 Approximation in a normed linear space
1.4 The L.-norms
1.5 A geometric view of best approximations
2 The uniqueness of best approximations
2.1 Convexity conditions
2.2 Conditions for the uniqueness of the best approximation
2.3 The continuity of best approximation operators
2.4 The 1-, 2- and 0o-norms
3 Approximation operators and some approximating functions
3.1 Approximation operators
3.2 Lebesgue constants
3.3 Polynomial approximations to diffcrentiable functions
3.4 Piecewise polynomial approximations
4 Polynomial interpolation
4.1 The Lagrange interpolation formula
4.2 The error in polynomial interpolation
4.3 The Chebyshev interpolation points
4.4 The norm of the Lagrange interpolation operator
5 Divided differences
5.1 Basic properties of divided differences
5.2 Newton’’s interpolation method
5.3 The recurrence relation for divided differences
5.4 Discussion of formulae for polynomial interpolation
5.5 Hermite interpolation
6 The uniform convergence of polynomial approximations
6.1 The Weierstrass theorem
6.2 Monotone operators
6.3 The Bernstein operator
6.4 The derivatives of the Bernstein approximations
7 The theory of minimax approximation
7.1 Introduction to minimax approximation
7.2 The reduction of the error of a trial approximation
7.3 The characterization theorem and the Haar condition
7.4 Uniqueness and bounds on the minimax error
8 The exchange algorithm
8.1 Summary of the exchange algorithm
8.2 Adjustment of the reference
8.3 An example of the iterations of the exchange algorithm
8.4 Applications of Chebyshev polynomials to minimax approximation
8.5 Minimax approximation on a discrete point set
9 The convergence of the exchange algorithm
9.1 The increase in the levelled reference error
9.2 Proof of convergence
9.3 Properties of the point that is brought into reference
9.4 Second-order convergence
10 Rational approximation by the exchange algorithm
10.1 Best minimax rational approximation
10.2 The best approximation on a reference
10.3 Some convergence properties of the exchange algorithm
10.4 Methods based on linear programming
11 Least squares approximation
11.1 The general form of a linear least squares calculation
11.2 The least squares characterization theorem
11.3 Methods of calculation
11.4 The recurrence relation for orthogonal polynomials
12 Properties of orthogonal polynomials
12.1 Elementary properties
12.2 Gaussian quadrature
12.3 The characterization of orthogonal polynomials
12.4 The operator R.
13 Approximation to periodic functions
13.1 Trigonometric polynomials
13.2 The Fourier series operator S.
13.3 The discrete Fourier series operator
13.4 Fast Fourier transforms
14 The theory of best L1 approximation
14.1 Introduction to best L1 approximation
14.2 The characterization theorem
14.3 Consequences of the Haar condition
14.4 The L1 interpolation points for algebraic polynomials
15 An example of L1 approximation and the discrete case
15.1 A useful example of L1 approximation
15.2 Jackson’’s first theorem
15.3 Discrete L1 approximation
15.4 Linear programming methods
16 The order of convergence of polynomial approximations
16.1 Approximations to non-differentiable functions
16.2 The Dini-Lipschitz theorem
16.3 Some bounds that depend on higher derivatives
16.4 Extensions to algebraic polynomials
17 The uniform boundedness theorem
17.1 Preliminary results
17.2 Tests for uniform convergence
17.3 Application to trigonometric polynomials
17.4 Application to algebraic polynomials
18 Interpolation by plecewise polynomials
18.1 Local interpolation methods
18.2 Cubic spline interpolation
18.3 End conditions for cubic spline interpolation
18.4 Interpolating splines of other degrees
19 B-splines
19.1 The parameters of a spline function
19.2 The form of B-splines
19.3 B-splines as basis functions
19.4 A recurrence relation for B-splines
19.5 The Schoenberg-Whitney theorem
20 Convergence properties of spline approximations
20.1 Uniform convergence
20.2 The order of convergence when f is differentiable
20.3 Local spline interpolation
20.4 Cubic splines with constant knot spacing
21 Knot positions and the calculation of spline approximation
21.1 The distribution of knots at a singularity
21.2 Interpolation for general knots
21.3 The approximation of functions to prescribed accurac
22 The Peano kernel theorem
22.1 The error of a formula for the solution of differential equations
22.2 The Peano kernel theorem
22.3 Application to divided differences and to polynomial interpolation
22.4 Application to cubic spline interpolation
23 Natural and perfect splines
23.1 A variational problem
23.2 Properties of natural splines
23.3 Perfect splines
24 Optimal interpolation
24.1 The optimal interpolation problem
24.2 L1 approximation by B-splines
24.3 Properties of optimal interpolation
Appendix A The Haar condition
Appendix B Related work and references
Index
