Definitions / 1: |
Preliminary remarks / 1.1: |
Trigonometric definitions and recurrences / 1.2: |
The first-kind polynomial T[subscript n] / 1.2.1: |
The second-kind polynomial U[subscript n] / 1.2.2: |
The third- and fourth-kind polynomials V[subscript n] and W[subscript n] (the airfoil polynomials) / 1.2.3: |
Connections between the four kinds of polynomial / 1.2.4: |
Shifted Chebyshev polynomials / 1.3: |
The shifted polynomials T*[subscript n], U*[subscript n], V*[subscript n], W*[subscript n] / 1.3.1: |
Chebyshev polynomials for the general range [a, b] / 1.3.2: |
Chebyshev polynomials of a complex variable / 1.4: |
Conformal mapping of a circle to and from an ellipse / 1.4.1: |
Chebyshev polynomials in z / 1.4.2: |
Shabat polynomials / 1.4.3: |
Problems for Chapter 1 / 1.5: |
Basic Properties and Formulae / 2: |
Introduction / 2.1: |
Chebyshev polynomial zeros and extrema / 2.2: |
Relations between Chebyshev polynomials and powers of x / 2.3: |
Powers of x in terms of {T[subscript n](x)} / 2.3.1: |
T[subscript n](x) in terms of powers of x / 2.3.2: |
Ratios of coefficients in T[subscript n](x) / 2.3.3: |
Evaluation of Chebyshev sums, products, integrals and derivatives / 2.4: |
Evaluation of a Chebyshev sum / 2.4.1: |
Stability of the evaluation of a Chebyshev sum / 2.4.2: |
Evaluation of a product / 2.4.3: |
Evaluation of an integral / 2.4.4: |
Evaluation of a derivative / 2.4.5: |
Problems for Chapter 2 / 2.5: |
The Minimax Property and Its Applications / 3: |
Approximation--theory and structure / 3.1: |
The approximation problem / 3.1.1: |
Best and minimax approximation / 3.2: |
The minimax property of the Chebyshev polynomials / 3.3: |
Weighted Chebyshev polynomials of second, third and fourth kinds / 3.3.1: |
The Chebyshev semi-iterative method for linear equations / 3.4: |
Telescoping procedures for power series / 3.5: |
Shifted Chebyshev polynomials on [0, 1] / 3.5.1: |
Implementation of efficient algorithms / 3.5.2: |
The tau method for series and rational functions / 3.6: |
The extended tau method / 3.6.1: |
Problems for Chapter 3 / 3.7: |
Orthogonality and Least-Squares Approximation / 4: |
Introduction--from minimax to least squares / 4.1: |
Orthogonality of Chebyshev polynomials / 4.2: |
Orthogonal polynomials and weight functions / 4.2.1: |
Chebyshev polynomials as orthogonal polynomials / 4.2.2: |
Orthogonal polynomials and best L[subscript 2] approximations / 4.3: |
Orthogonal polynomial expansions / 4.3.1: |
Convergence in L[subscript 2] of orthogonal expansions / 4.3.2: |
Recurrence relations / 4.4: |
Rodrigues' formulae and differential equations / 4.5: |
Discrete orthogonality of Chebyshev polynomials / 4.6: |
First-kind polynomials / 4.6.1: |
Second-kind polynomials / 4.6.2: |
Third- and fourth-kind polynomials / 4.6.3: |
Discrete Chebyshev transforms and the fast Fourier transform / 4.7: |
The fast Fourier transform / 4.7.1: |
Discrete data fitting by orthogonal polynomials: the Forsythe-Clenshaw method / 4.8: |
Bivariate discrete data fitting on or near a family of lines or curves / 4.8.1: |
Orthogonality in the complex plane / 4.9: |
Problems for Chapter 4 / 4.10: |
Chebyshev Series / 5: |
Introduction--Chebyshev series and other expansions / 5.1: |
Some explicit Chebyshev series expansions / 5.2: |
Generating functions / 5.2.1: |
Approximate series expansions / 5.2.2: |
Fourier-Chebyshev series and Fourier theory / 5.3: |
L[subscript 2]-convergence / 5.3.1: |
Pointwise and uniform convergence / 5.3.2: |
Bivariate and multivariate Chebyshev series expansions / 5.3.3: |
Projections and near-best approximations / 5.4: |
Near-minimax approximation by a Chebyshev series / 5.5: |
Equality of the norm to [lambda][subscript n] / 5.5.1: |
Comparison of Chebyshev and other orthogonal polynomial expansions / 5.6: |
The error of a truncated Chebyshev expansion / 5.7: |
Series of second-, third- and fourth-kind polynomials / 5.8: |
Series of second-kind polynomials / 5.8.1: |
Series of third-kind polynomials / 5.8.2: |
Multivariate Chebyshev series / 5.8.3: |
Lacunary Chebyshev series / 5.9: |
Chebyshev series in the complex domain / 5.10: |
Chebyshev-Pade approximations / 5.10.1: |
Problems for Chapter 5 / 5.11: |
Chebyshev Interpolation / 6: |
Polynomial interpolation / 6.1: |
Orthogonal interpolation / 6.2: |
Chebyshev interpolation formulae / 6.3: |
Aliasing / 6.3.1: |
Second-kind interpolation / 6.3.2: |
Third- and fourth-kind interpolation / 6.3.3: |
Conditioning / 6.3.4: |
Best L[subscript 1] approximation by Chebyshev interpolation / 6.4: |
Near-minimax approximation by Chebyshev interpolation / 6.5: |
Problems for Chapter 6 / 6.6: |
Near-Best L[subscript [infinity]], L[subscript 1] and L[subscript p] Approximations / 7: |
Near-best L[subscript [infinity]] (near-minimax) approximations / 7.1: |
Second-kind expansions in L[subscript [infinity]] / 7.1.1: |
Third-kind expansions in L[subscript [infinity]] / 7.1.2: |
Near-best L[subscript 1] approximations / 7.2: |
Best and near-best L[subscript p] approximations / 7.3: |
Complex variable results for elliptic-type regions / 7.3.1: |
Problems for Chapter 7 / 7.4: |
Integration Using Chebyshev Polynomials / 8: |
Indefinite integration with Chebyshev series / 8.1: |
Gauss-Chebyshev quadrature / 8.2: |
Quadrature methods of Clenshaw-Curtis type / 8.3: |
First-kind formulae / 8.3.1: |
Second-kind formulae / 8.3.3: |
Third-kind formulae / 8.3.4: |
General remark on methods of Clenshaw-Curtis type / 8.3.5: |
Error estimation for Clenshaw-Curtis methods / 8.4: |
Fitting an exponential curve / 8.4.1: |
Other abscissae and polynomials / 8.4.3: |
Some other work on Clenshaw-Curtis methods / 8.5: |
Problems for Chapter 8 / 8.6: |
Solution of Integral Equations / 9: |
Fredholm equations of the second kind / 9.1: |
Fredholm equations of the third kind / 9.3: |
Fredholm equations of the first kind / 9.4: |
Singular kernels / 9.5: |
Hilbert-type kernels and related kernels / 9.5.1: |
Symm's integral equation / 9.5.2: |
Regularisation of integral equations / 9.6: |
Discrete data with second derivative regularisation / 9.6.1: |
Details of a smoothing algorithm (second derivative regularisation) / 9.6.2: |
A smoothing algorithm with weighted function regularisation / 9.6.3: |
Evaluation of V ([lambda]) / 9.6.4: |
Other basis functions / 9.6.5: |
Partial differential equations and boundary integral equation methods / 9.7: |
A hypersingular integral equation derived from a mixed boundary value problem for Laplace's equation / 9.7.1: |
Problems for Chapter 9 / 9.8: |
Solution of Ordinary Differential Equations / 10: |
A simple example / 10.1: |
Collocation methods / 10.2.1: |
Error of the collocation method / 10.2.2: |
Projection (tau) methods / 10.2.3: |
Error of the preceding projection method / 10.2.4: |
The original Lanczos tau ([tau]) method / 10.3: |
A more general linear equation / 10.4: |
Collocation method / 10.4.1: |
Projection method / 10.4.2: |
Pseudospectral methods--another form of collocation / 10.5: |
Definitions / 1: |
Preliminary remarks / 1.1: |
Trigonometric definitions and recurrences / 1.2: |
The first-kind polynomial T[subscript n] / 1.2.1: |
The second-kind polynomial U[subscript n] / 1.2.2: |
The third- and fourth-kind polynomials V[subscript n] and W[subscript n] (the airfoil polynomials) / 1.2.3: |