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